Maëlys

William, Baptiste, Matthias Joachim Valentin

Camille Trang

Remerciements

Mes premiers remerciements se tournent d'abord vers ma direction de thèse, Manon et Patrick, qui m'ont proposé ces sujets de travail passionnants, et ont encadré mon stage et ma thèse. Merci pour m'avoir accompagnée dans ces premiers pas de la recherche, pour votre patience, votre bienveillance et vos conseils. Merci Patrick, pour m'avoir acceptée dans tes derniers thésards. Merci Manon, pour m'avoir acceptée comme première doctorante et m'avoir intégrée dans la communauté math-bio. Merci à tous les deux pour nos échanges, pour m'avoir expliqué maintes choses sur l'histoire et le fonctionnement de la recherche (et pour m'avoir montré qu'on peut, et qu'on doit, râler quand quelque chose ne va pas.) Je me dois de remercier également mon jury. Eva Löcherbach et François Bolley, merci pour votre relecture attentive de mon manuscrit et vos commentaires. Christophe Poquet et Gersende Fort, je vous suis très reconnaissante d'avoir accepté de faire partie de mon jury (paritaire d'ailleurs, c'est assez rare pour le noter). Eva, je te remercie en particulier de m'avoir accueillie avec un grand sourire dans la communauté mathématique, et de m'avoir invitée à présenter à Paris. François, Christophe, je vous remercie tous les deux très chaleureusement pour m'avoir aidée et encouragée dans les candidatures de post-doctorat. Je reviendrai peut-être vers vous pour la prochaine campagne ! Gersende, comme tous les chercheurs du laboratoire, je vous remercie pour toutes les discussions que nous avons pu avoir. Merci pour avoir répondu à mes questions et à ma curiosité sur le fonctionnement du monde de la recherche et de l'université. De manière générale, j'ai eu la chance de rencontrer, au laboratoire comme à l'extérieur, des chercheurs et des chercheuses toujours ouverts à la discussion. Je souhaite également remercier Arnaud Guillin pour nos conversations mathématiques très intéressantes. Merci pour nous avoir accompagné Pierre et moi dans le projet sur les FitzHugh-Nagumo. Pierre, que je remercie également : ça a été un plaisir de travailler avec toi, et ce sera à nouveau avec plaisir que nous pourrons collaborer. Merci Sylvie Roeilly pour ton accueil à Potsdam et pour ton aide dans la recherche de post-doctorat. Merci également pour ton invitation à passer quelques semaines ou mois : le covid s'en est mêlé, mais nous nirons par trouver le moment ! Et à la suite de tous ces chercheurs et toutes ces chercheuses, je glisse un mot pour Alexandra Asanovna Elbakyan, ainsi que toutes les autres personnes s'attachant à la science en accès libre. Mon doctorat n'aurait pas été le même sans mon expérience à

Chapitre 1 Introduction

Cette thèse porte sur deux processus stochastiques, les processus de Hawkes et les processus de FitzHugh-Nagumo. Ceux-ci, particulièrement distincts comme nous le verrons dans la suite, peuvent tous deux servir à de la modélisation. Là où les processus de Hawkes sont applicables à des phénomènes très divers, comme les séismes [START_REF] Hawkes | Spectra of Some Self-Exciting and Mutually Exciting Point Processes[END_REF][START_REF] Hawkes | A Cluster Process Representation of a Self-Exciting Process[END_REF], les réseaux sociaux [START_REF] Fox | Modeling E-mail Networks and Inferring Leadership Using Self-Exciting Point Processes[END_REF][START_REF] Fang | Group Network Hawkes Process[END_REF], la nance [START_REF] Hawkes | Hawkes Processes and Their Applications to Finance: A Review[END_REF][START_REF] Bacry | Hawkes Model for Price and Trades High-Frequency Dynamics[END_REF], la biologie (génomique [START_REF] Reynaud | Adaptive Estimation for Hawkes Processes; Application to Genome Analysis[END_REF], neurologie [START_REF] Löcherbach | Spiking Neurons : Interacting Hawkes Processes, Mean Field Limits and Oscillations[END_REF][START_REF] Richard Hansen | Lasso and Probabilistic Inequalities for Multivariate Point Processes[END_REF]) ; les processus de FitzHugh-Nagumo ont été spéciquement construits pour une application : les décharges électriques des neurones. Nous commençons donc par introduire, en quelques pages, la modélisation des neurones dans le cerveau, an de comprendre comment deux processus si distincts peuvent être intéressants dans ce cadre.

Ensuite nous décrirons plus explicitement ces processus, le contexte dans lequel nous les avons étudiés et les résultats obtenus.

1.1 Quelques explications sur les neurones 1.1.1 Préambule biologique Le fonctionnement biologique du cerveau et des neurones est encore très peu connu. Diérentes approches, et diérents modèles existent an de, petit à petit, comprendre certaines spécicités et certains fonctionnements. Notre point de vue ici, via les modèles que nous étudions, est un point de vue microscopique : l'idée est d'essayer de comprendre comment fonctionne un neurone, puis plusieurs neurones pour comprendre le fonctionnement général d'un ensemble de neurones. Il ne s'agit bien sûr pas de modéliser un cerveau entier : un humain a de l'ordre de 10 11 neurones, ce qui est immense, et ne peut donc pas être étudié en considérant les neurones un à un. D'autres types de modèles, dits macroscopiques, existent pour ces questions. Des liens existent entre modèles microscopiques et modèles macroscopiques, notamment via ce que nous avons étudié dans un cadre particulier avec la propagation du chaos. Tout d'abord, un neurone est une cellule. Cette dernière est spécique et permet de transmettre des signaux électriques sur des distances pouvant être grandes.

Il est composé d'un soma (le corps de la cellule), d'un axone et d'un ou plusieurs dendrites, comme le décrit la gure 1.1. Les dendrites permettent de réceptionner les signaux émis par d'autres neurones. L'axone, unique lui, a plusieurs terminaisons axonales qui permettent au neurone de transmettre un signal. Un neurone typique reçoit des signaux venant de plusieurs milliers de neurones. Chacune de ces contributions entraîne un changement du potentiel de membrane du neurone. Ces apports peuvent être d'amplitude variée et entraînent un changement de potentiel proportionnel. Le neurone va ensuite générer une décharge électrique qui est, par la suite, propagée le long de son axone en direction des autres neurones.

Les déclenchements de décharges ne sont pas encore entièrement compris. L'idée classique est de considérer un seuil : tant que le potentiel du neurone est en dessous du seuil, il ne se passe rien, et lorsqu'il le dépasse, une décharge est produite avec la totalité du potentiel. Cette idée est mise à mal par plusieurs expériences. En eet, à plusieurs reprises, des biologistes ont tenté de déterminer le seuil de déclenchement pour certains types de neurones, mais ils n'ont pas pu l'identier. En fait, selon la durée du signal reçu par le neurone (instantané, ou sur une durée plus longue), son type (signal unique ou répétitif ), sa période (dans le cas où il est périodique), le seuil de déclenchement peut être diérent. Il ne s'agit pas simplement d'une condition en 1. Nicolas Rougier (2007) Wikimedia Commons. (https://commons.wikimedia.org/wiki/ 1.1. Quelques explications sur les neurones 3 tout-ou-rien : "au-delà de telle valeur" et "en dessous de telle valeur".

Diérents modèles mathématiques ont donc été construits pour étudier la transmission de ces signaux électriques entre les neurones. Parmi ces modèles, certains considèrent un seuil, et d'autres ont des interactions plus nuancées.

Les questions des biologistes sont extrêmement variées. Ainsi, le choix du modèle dépend des points d'intérêts, du type de données fournies et de la précision souhaitée.

Les données obtenues sont souvent de la forme de spiking train, c'est-à-dire d'une suite de temps de décharge de neurones. Il n'est d'ailleurs pas toujours possible d'isoler les temps de décharge pour un neurone par rapport aux autres d'un point de vue expérimental, et des outils statistiques sont donc développés pour le faire.

Le taux de décharge, ou ring rate, est une quantité souvent étudiée dans les articles.

Il s'agit de l'inverse du nombre de décharges sur une certaine durée, qui permet de calibrer le modèle à partir de vraies données mais qui est également une quantité intéressante en soi.

Les modèles

Lorsque l'on s'intéresse aux comportements des neurones, deux types de modèles existent : les modèles à sauts, dans lesquels on s'intéresse aux instants de décharge du neurone (et éventuellement à leur amplitude), et les modèles continus, dans lesquels on considère l'évolution du potentiel du neurone.

La construction et l'utilisation de ces deux types de modèles dans les neurosciences ont été faites de manière diérente : là où les modèles continus ont en général été construits à partir des expériences et d'hypothèses, et sont spéciques aux neurones ; les modèles à sauts ont pu être construits et étudiés pour d'autres objectifs, puis ont été appliqués aux neurones.

Modèles à sauts

Lorsque l'on regarde une suite d'instants aléatoires, le plus naturel est de considérer un processus de saut. Ces processus, aussi appelés processus ponctuels sur R, sont représentés par une suite de temps (S i ) i∈N ou de manière équivalente par une suite d'intervalles (τ i ) i∈N * = (S i -S i-1 ) i∈N * . Ils sont caractérisés par une valeur initiale S 0 , souvent nulle, et par une loi pour les τ i .

Le processus le plus simple est bien entendu le processus de Poisson homogène : la suite (τ i ) i∈N * est i.i.d. et τ 1 suit une loi exponentielle de paramètre constant λ. On dit alors que l'intensité du processus Λ est constante et vaut λ. Dans le cadre de modélisation de neurones, ce processus est trop simple pour être utilisé : par exemple, il ne permet pas de représenter des périodes réfractaires (périodes qui suivent un déclenchement de décharge et pendant lesquelles le neurone ne peut plus se redéclencher). Une seconde limite, assez générale pour les processus ponctuels, est que le processus de Poisson homogène est stationnaire, et il a déjà été montré, par exemple dans [START_REF] Shinomoto | Estimating the Firing Rate[END_REF], que les décharges neuronales ne peuvent pas être entièrement modélisées par un processus stationnaire.

Une manière de complexier cette idée est de considérer les processus de renouvellement, comme dans [START_REF] Pipa | Impact of Spike Train Autostructure on Probability Distribution of Joint Spike Events[END_REF]. Ces processus sont décrits par une suite (τ i ) i∈N * qui est i.i.d. mais dont la loi n'est pas nécessairement exponentielle. Dans [START_REF] Pipa | Impact of Spike Train Autostructure on Probability Distribution of Joint Spike Events[END_REF], on suppose que τ 1 suit une loi Gamma ou une loi Log-Normale. Cependant ce modèle présente la même limite que le modèle de Poisson homogène : le processus est stationnaire. Il s'agit néanmoins d'un modèle très simple qui peut être utilisé pour certaines études.

Pour aller plus loin, [START_REF] Pouzat | Automatic Spike Train Analysis and Report Generation. An Implementation with R, R2HTML and STAR[END_REF] et [START_REF] Chevallier | Microscopic Approach of a Time Elapsed Neural Model[END_REF] ont considéré des processus de Wold, c'est-à-dire des processus dont le futur saut dépend du dernier instant de saut, mais également de la longueur du dernier intervalle : en fait τ n+1 dépend de τ n .

Une seconde manière de complexier l'idée d'un processus de Poisson homogène est de s'intéresser à des processus de Poisson inhomogènes, dont l'intensité Λ varie au cours du temps. C'est le type de processus considéré dans [START_REF] Ventura | Statistical Analysis of Temporal Evolution in Single-Neuron Firing Rates[END_REF] et dans [START_REF] Reynaud-Bouret | Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis[END_REF]. Néanmoins, [START_REF] Grün | Unitary Joint Events in Multiple Neuron Spiking Activity: Detection, Signicance, and Interpretation[END_REF] et [START_REF] Pipa | Signicance of Joint-Spike Events Based on Trial-Shuing by Ecient Combinatorial Methods[END_REF] ont mis en évidence une dépendance des décharges des neurones entre elles, ce qui ne peut pas être modélisé par ces processus. Les processus de Hawkes permettent d'étendre la notion de processus de Poisson en considérant une dépendance des évènements entre eux, c'est pourquoi [START_REF] Reynaud-Bouret | Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis[END_REF] et [START_REF] Chevallier | Microscopic Approach of a Time Elapsed Neural Model[END_REF] s'y sont intéressés dans le cadre de la modélisation de neurones. Même si les processus de Hawkes sont également stationnaires, ils permettent tout de même de modéliser dans une certaine mesure des décharges neuronales. Ce processus que nous avons choisi d'étudier est déni plus en détail dans la suite.

Modèle continu

Pour étudier l'évolution du potentiel d'un neurone, diérents modèles ont été décrits avec des équations diérentielles ordinaires (EDO). Ici, nous n'en évoquerons que quelques-uns, en décrivant en particulier les équations diérentielles stochastiques (EDS) qui en découlent. Pour avoir plus d'informations et d'explications biochimiques sur les modèles, le livre [START_REF] Izhikevich | Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting[END_REF] présente les diérents systèmes dynamiques (déterministes) utilisés en neuroscience.

De manière générale, les systèmes que nous présentons ici découlent tous de la représentation d'une synapse comme d'un circuit électrique, et de En particulier, une version stochastique de ce modèle existe et est étudié, notamment dans [START_REF] Jahn | Motoneuron Membrane Potentials Follow a Time Inhomogeneous Jump Diusion Process[END_REF]. Dans cet article, les paramètres sont estimés à partir des données expérimentales, et le modèle semble bien adapté.

Néanmoins, ce modèle est descriptif et ne permet pas de comprendre la manière dont les décharges se produisent. Hodgkin et Huxley [START_REF] Hodgkin | A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve[END_REF], en 1952, ont travaillé sur l'axone géant du calmar pour comprendre plus spéciquement le fonctionnement du neurone. Ils ont décrit le courant du modèle en le décomposant en trois courants ioniques : I model = I Na + I K + I l . Les ions sodium et potassium représentent la part principale de ce courant, et I l est le courant "de fuite", généré par les ions chlorure et d'autres ions. I K et I Na sont décrits à partir de X et de trois variables n, m et h qui sont les variables d'activation ou de désactivation liées aux canaux ioniques.

Pour chacune de ces variables, une EDO permet de décrire son comportement (en fonction d'elle-même et de X). Ce modèle est donc constitué de quatre EDO pour un seul neurone. Diérentes versions stochastiques de ce modèle ont été étudiées.

Ainsi, [START_REF] Höpfner | Strongly Degenerate Time Inhomogeneous SDEs: Densities and Support Properties. Application to HodgkinHuxley Type Systems[END_REF] considère un courant d'entrée I input stochastique, tandis que [4] considère un bruit brownien sur les diérentes équations. Ce dernier article, consacré à l'étude de ce modèle dans le cadre de plusieurs neurones en interaction champ-moyen, est complété par [START_REF] Bossy | Clarication and Complement to Mean-Field Description and Propagation of Chaos in Networks of HodgkinHuxley and FitzHughNagumo Neurons[END_REF].

Le modèle de Hodgkin-Huxley est reconnu comme étant très complet, mais en raison de sa complexité (chaque neurone est décrit par 4 EDO ou EDS non-linéaires), plusieurs modèles simpliés ont été proposés et étudiés, en règle générale en se réduisant à 2 EDO ou EDS. Certains modèles se sont concentrés sur les courants de certains ions, comme le modèle de Morris-Lecar [START_REF] Morris | Voltage Oscillations in the Barnacle Giant Muscle Fiber[END_REF], aussi appelé modèle I Ca + I K .

Une version stochastique a été décrite et [START_REF] Ditlevsen | Estimation in the Partially Observed Stochastic MorrisLecar Neuronal Model with Particle Filter and Stochastic Approximation Methods[END_REF] décrit une méthode pour estimer des paramètres physiologiques à partir de ce modèle et de données partielles obtenues par l'expérience.

Une autre simplication possible de ce modèle est le modèle de FitzHugh-Nagumo, sur lequel nous nous sommes concentrés dans cette thèse. Celui-ci a été déni en 1961 par FitzHugh [START_REF] Fitzhugh | Impulses and Physiological States in Theoretical Models of Nerve Membrane[END_REF] et le circuit électrique équivalent a été décrit par Nagumo, Arimoto et Yoshizawa en 1962 [START_REF] Nagumo | An Active Pulse Transmission Line Simulating Nerve Axon[END_REF].

1.2 Plan de la thèse et chaque chapitre Pendant cette thèse, je me suis concentrée sur les deux modèles précédemment cités : le modèle de Hawkes et le modèle de FitzHugh-Nagumo.

Tout d'abord, j'ai souhaité généraliser des résultats connus pour une sous-catégorie de processus de Hawkes : le cas des processus dit linéaires et auto-excitants. Je me suis intéressée à des processus qui pouvaient représenter de l'auto-inhibition ou une auto-inuence à la fois inhibitrice et excitatrice. L'objectif est de permettre la modélisation de phénomènes plus complexes : il est par exemple connu que les neurones peuvent être inhibés. On peut également considérer une auto-inuence mixte, d'abord inhibante -ce qui empêche l'arrivée d'un second évènement trop tôt après un premier -puis excitante après une certaine durée. Je me suis restreinte au cas où cette auto-inuence est sur une durée limitée. Cette restriction est cohérente avec les applications possibles, et permet de représenter les processus de Hawkes comme des processus cumulatifs, qui sont un type de processus de renouvellement plus général.

Cette représentation m'a permis de démontrer une loi des grands nombres et un théorème central limite, qui était déjà connus pour les processus cumulatifs, dans mon cadre. Le Chapitre 3 décrit ce travail.

J'ai souhaité compléter mon étude avec un Principe de Grandes Déviations, ou a minima des inégalités de grandes déviations. Néanmoins, ce principe n'existait pas pour les processus cumulatifs dans la littérature dans le cadre où je souhaitais l'appliquer. J'ai donc démontré un Principe de Grandes Déviations, et des inégalités de grandes déviations sous certaines hypothèses : cette étude est détaillée dans le Chapitre 2. J'ai ensuite pu appliquer ce Théorème dans le cadre des processus de

Hawkes, ce qui achève le Chapitre 3.

Pour cette raison, je présente de manière liée ces deux Chapitres dans la Sous-Section 1.2.1 suivante.

Ensuite, je me suis intéressée au modèle de FitzHugh-Nagumo stochastique.

Alors que, pour les processus de Hawkes, je me focalisais sur un seul processus, ici j'ai souhaité considérer des processus en interaction, pouvant représenter plusieurs neurones qui communiquent. Je me suis concentrée sur le cadre d'une interaction champ-moyen, et j'ai montré une propagation du chaos, d'abord non-uniforme en temps, puis dans un second temps, uniforme en temps. Pour ce faire, j'ai collaboré avec Pierre Le Bris, et nous avons utilisé une méthode de couplage mixte, c'est-àdire un couplage synchrone sur un certain sous-espace, et un couplage symétrique sur l'espace complémentaire. Cette approche est l'objet du Chapitre 4.

1.2.1 Processus à sauts -Chapitres 2 et 3

Dénitions et propriétés des processus cumulatifs

Les processus cumulatifs (cumulative process) ont été décrits par Smith [START_REF] Walter | Regenerative Stochastic Processes[END_REF] et sont appliqués dans de nombreux contextes, comme en nance où ils sont appelés processus de renouvellement composé (compound-renewal processes) ou processus de renouvellement à récompense (renewal-reward processes). Ces processus cumulent des variables aléatoires indépendantes au cours du temps (continu). Ces variables aléatoires sont ajoutées sur des intervalles de temps donnés par un processus de 1.2. Plan de la thèse et chaque chapitre 7 renouvellement.

Nous considérons un espace de probabilité muni d'une ltration appropriée

(Ω, F, (F t ) t≥0 , P) qui satisfait les hypothèses habituelles.

Dénition 1.2.1. Un processus (Z t ) t : R + → R est dit cumulatif s'il vérie les propriétés suivantes :

1. Z 0 = 0, 2. il existe un processus de renouvellement (S n ) n tel que pour tout n, (Z Sn+t -Z Sn ) t≥0 est indépendant de S 0 , ...S n et de (Z s ) s<Sn , 3. la distribution de (Z Sn+t -Z Sn ) t≥0 est indépendante de n.

Néanmoins, travailler avec cette dénition n'est pas toujours simple. On préfère en général travailler avec l'écriture

Z t = W 0 (t) + Mt i=1 W i + r t , (1.2.1) où :
(τ i ) i est la suite de durées associée au processus de renouvellement (S n ) n ; M t = sup {n ≥ 0, S n ≤ t}, le processus de comptage associé à (S n ) n ; W i = Z Sn -Z S n-1 ; W 0 (t) = Z t∧τ 1 = Z inf(t,τ 1 ) ; r t = Z t -Z Mt . En particulier, on a alors (τ 1 , W 1 ), (τ 2 , W 2 ), . . . une suite i.i.d. de couples de variables aléatoires dans R + × R.

Remarque 1.2.2. On peut dénir un exemple de processus cumulatif de la façon suivante : soit une suite i.i.d. de couples de variables aléatoires (τ 1 , W 1 ), (τ 2 , W 2 ), . . . dans R + × R, dénies sur un espace de probabilité (Ω, P). Nous considérons le processus de renouvellement associé à (τ i ) i≥1

S 0 = 0, S n = n k=1 τ k , et M t = sup {n ≥ 0, S n ≤ t} . La variable aléatoire Z t = Mt i=1
W i est un processus cumulatif, avec W 0 = 0 et r t = 0.

Il est important de noter qu'en général, à i xé, τ i et W i ne sont pas indépendants. Comme les couples (τ i , W i ) i sont i.i.d., on supprimera dans la suite l'indice i lorsque l'on traitera de la distribution (et des quantités associées comme l'espérance et la variance) de (τ i , W i ) et nous écrirons simplement (τ, W ).

Z t -t E[W ] E[τ ] √ t -→ t→∞ N 0, σ 2 où σ 2 = 1 E(τ ) Var W - E[W ] E[τ ]
τ peuvent être trouvés dans le livre d'Asmussen [START_REF] Asmussen | Applied Probability and Queues. 2nd ed. Stochastic Modelling and Applied Probability[END_REF], théorème 3.1 et théorème 3.2.

L'objectif de notre travail est d'établir un principe de grande déviation pour les processus cumulatifs. Certaines études ont déjà été eectuées dans ce sens. En particulier, Duy et Metcalfe [START_REF] Duy | How to Estimate the Rate Function of a Cumulative Process[END_REF] ont travaillé sur l'estimation d'une fonction de taux pour un processus cumulatif (s'il admet un principe de grande déviation). Dans une suite d'articles, Borovkov et Mogulskii ([9,[START_REF] Borovkov | Large Deviation Principles for Trajectories of Compound Renewal Processes[END_REF][START_REF] Borovkov | Large Deviation Principles for Trajectories of Compound Renewal Processes[END_REF]) ont également étudié le principe de grande déviation, sous certaines hypothèses de comparaison entre la loi jointe de (τ, W ) et la loi de τ . Certains détails ne nous semblaient pas clairs, et nous avons souhaité établir un principe dans un cadre diérent, l'objectif étant de pouvoir l'appliquer aux processus de Hawkes.

Lefevere, Mariani et Zambotti [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] ont travaillé sur des processus cumulatifs spéciques pour lesquels W i = F (τ i ), pour F une fonction déterministe qui est supposée positive, bornée et continue. Nous avons choisi de suivre leur approche.

1.2.1.2 Chapitre 2 : Grandes déviations pour les processus cumulatifs Principe de grandes déviations -rappels généraux L'objectif de ce chapitre est de prouver un Principe de Grandes Déviations pour le processus Z t /t. Rappelons quelques dénitions sur les grandes déviations (tirées de [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]).

Une famille de mesures de probabilité (η t ) t∈R + sur un espace topologique (X , T X ) satisfait un principe de grandes déviations avec la fonction de taux J(.) et la vitesse γ(t) = t si J : X → [0, +∞] est semi-continue inférieurement et satisfait Puisque la fonction de taux J est semi-continue inférieurement, les ensembles de niveau {X, J(x) ≤ a} sont fermés. S'ils sont également compacts, alors J sera qualiée de bonne fonction de taux.

-inf x∈O J(x) ≤ lim
Nous considérerons ici uniquement la vitesse γ(t) = t, nous n'y ferons donc plus référence.

Une notion particulièrement importante pour notre objectif est la notion d'approximation exponentiellement bonne. alors Y t satisfait un Principe de Grandes Déviations complet, avec la fonction de taux J.

Le premier et le dernier de ces points sont contenus dans le Théorème 4.2.16 de [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. Le second point est une conséquence du fait que les boules fermées sont des compacts dans R k .

Habituellement, ce Théorème est susant pour prouver un Principe de Grandes

Résultats de Grandes Déviations

Nous faisons les hypothèses suivantes :

Hypothèse 1.2.6. J n (z) .

(

Stratégie de preuve

La preuve de ce théorème est divisée en plusieurs étapes. Tout d'abord, nous commençons par réduire le problème aux W bornés, en remplaçant W par W n , puis aux W à valeurs dans un ensemble ni. Sous de bonnes hypothèses, ces diérents processus cumulatifs sont des approximations exponentiellement bonnes, et le Théorème 1.2.4 permettra de conclure. Il nous sut alors de nous concentrer sur le cas des W à valeurs dans un ensemble ni. En suivant la démarche de [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF], nous introduisons une mesure empirique µ n t asso- ciée au processus et nous prouvons le Principe de Grandes Déviations pour cette mesure :

Théorème 1.2.10. Soit µ n t la mesure empirique d'un processus cumulatif satisfaisant les Hypothèses 1.2.6, et tel que W n prenne ses valeurs dans {1, ..., n}. Alors la famille (P n t ) t≥0 des distributions de µ n t satisfait un principe de grande déviation avec une bonne fonction de taux I n quand t → ∞ avec une vitesse t, où I n n'est pas précisée ici, mais est dénie dans le corps du manuscrit, dans (2.4.5).

Pour prouver ce Théorème, nous étudions la fonction de taux de la mesure empirique. Il s'agit d'une étude assez technique. Nous adaptons les arguments de [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] en utilisant des conditionnements par rapport à chaque valeur de W . Le fait que W soit à valeurs dans un ensemble ni nous permet de conditionner sans diculté et nous permet d'utiliser des arguments de compacité puisque M 1 ({1, ..., n}) est compact.

De nombreux arguments sont proches de [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF], mais nous devions écrire plus en détail certaines preuves. Certains points topologiques devaient également être clariés.

Pour l'article qui a été soumis et que nous reprenons ici, nous avons fait le choix de faire référence aux résultats correspondants dans [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] seulement lorsqu'ils pouvaient être reproduits ligne à ligne dans notre cas. Nous avons ajouté en annexe les preuves venant de [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF].

Le reste de la preuve de ce Théorème est similaire au travail eectué par [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF].

Le principe de contraction pour les Principes de Grandes Déviations nous permet nalement de transférer ce principe de la mesure empirique au processus cumulatif approché (de cumulants bornés, et à valeurs dans un ensemble ni). Pour ce faire, nous avons dû travailler sur la fonction de taux associée an d'en extraire une écriture la plus claire possible.

Il est à noter que, si nous avons réussi à prouver un principe dit complet pour les processus cumulatifs approchés (de cumulants bornés, et à valeurs dans un ensemble ni), nous n'obtenons qu'un principe dit faible dans le cadre général. Nous nous restreignons avec des hypothèses sur la loi de W pour assurer un principe complet.

Chapitre 3 : Propriétés asymptotiques pour les processus de Hawkes Dénitions

Nous considérons un espace de probabilité muni d'une ltration appropriée (Ω, F, (F t ) t≥0 , P) qui satisfait les hypothèses habituelles. Dénition 1.2.11. Soit λ > 0 et h : (0, +∞) → R une fonction mesurable signée.

Soit N 0 un processus ponctuel, localement ni, sur (-∞, 0], qui suit la loi m. Le processus ponctuel N h sur R est un processus de Hawkes sur (0, +∞), de condition initiale N 0 et de mesure de reproduction µ(dt) = h(t)dt si :

N h | (-∞,0] = N 0 , la mesure d'intensité N h | (0,+∞) conditionnellement à (F t ) t≥0 est absolument
continue par rapport à la mesure de Lebesgue et a pour densité :

Λ h : t ∈ (0, +∞) → λ + (-∞,t) h(t -u)N h (du) + .
(1.2.12) où x + = max(x, 0).

On nomme h fonction de reproduction du processus, ou noyau de reproduction. Remarque 1.2.12. Il est possible de dénir un processus de Hawkes plus général, en choisissant

Λ h (t) = f (-∞,t) h(t -u)N h (du) ,
1.2. Plan de la thèse et chaque chapitre 13 avec f une fonction positive appelée fonction de taux de saut. Dans ce travail, nous traitons seulement le cas f : y → max(0, λ + y). On remarque que si h est une fonction positive, on peut remplacer f : y → max(0, λ+ y) par une fonction ane y → λ+y. On qualie ces processus de Hawkes de linéaire.

Ce sont les processus de Hawkes historiques, dénis par Oakes et Hawkes [START_REF] Hawkes | A Cluster Process Representation of a Self-Exciting Process[END_REF], qui ont été particulièrement étudiés. Nous utiliserons dans la suite certains résultats obtenus pour ceux-ci pour les étendre aux processus de Hawkes avec f : y → max(0, λ + y).

La proposition suivante donne une représentation explicite d'un processus de

Hawkes comme une solution d'une EDS dirigée par un mesure aléatoire de Poisson sur R 2 , et établit une propriété de couplage très importante. Proposition 1.2.13 (Proposition 2.1 de [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF]). Soit Q une mesure aléatoire de Poisson de dimension deux, sur (0, +∞) × (0, +∞), d'intensité unitaire dxdy et adaptée à (F t ) t≥0 . Nous considérons le système

   N h = N 0 + (0,+∞)×(0,+∞) δ u 1 θ≤Λ h (u) Q(du, dθ) Λ h (u) = λ + (-∞,u) h(u -s)N h (ds) + , u > 0, (1.2.13) où λ > 0 est un taux d'immigration, h : (0, +∞) → R est une fonction mesurable signée et N 0 est une condition initiale de loi m sur (-∞, 0].

Nous considérons le système similaire pour

N h + , dans lequel h est remplacé par h + (.) = max(h(.), 0). Nous supposons que h + 1 := h + L 1 (du) < 1 et que la distribution m satisfait : ∀t > 0, t 0 E m (-∞,0] h + (u -s)N 0 (ds) du < +∞. (1.2.
L(h) := sup{t > 0, |h(t)| > 0} < ∞. ii) h + 1 := +∞ 0 h + (u) du < 1, où h + (x) = max(h(x), 0).
iii) λ > 0, iv) la condition initiale sur ] -∞, 0[ ne contient aucun saut i.e. m = δ ∅ .

Nous nous intéressons au comportement asymptotique du nombre de sauts du processus N h sur l'intervalle [0, t] que nous notons :

N h t = N h ([0, t]),

Les processus de Hawkes sont des processus cumulatifs

Soit N h un processus de Hawkes, qui suit les Hypothèses 1.2.14, avec pour condition initiale N 0 = ∅. Ce processus de Hawkes peut être en fait vu comme un processus cumulatif.

On écrit U 1 , U 2 , U 3 , . . . ses sauts successifs.

Nous introduisons les temps de renouvellement du processus, qui permettent de diviser la demi-droite temporelle R + en fenêtres temporelles (ou intervalles) du processus, qui sont indépendantes et identiquement distribuées. Leurs durées sont données par τ 1 , τ 2 , . . . .

Commençons par dénir le temps d'arrêt

τ 1 = inf{t > U 1 , N h ((t -L(h), t]) = 0},
qui est le premier instant après U 1 tel qu'il n'y a pas eu de saut pendant une durée L(h). Nous posons également

S 0 = 0 et S 1 = τ 1 .

Plan de la thèse et chaque chapitre 15

Dénissons maintenant

W 1 = N h ([U 1 , S 1 ]) = N h ([0, S 1 ]),
le nombre de sauts du processus dans la première fenêtre temporelle. Nous renumérotons les instants de sauts dans cette première fenêtre temporelle :

U 1 j = U j , ∀j ∈ {1, . . . , W 1 }.
Nous verrons par la suite que τ 1 et W 1 sont nis presque sûrement. Par récursivité, soit i ∈ N * tel que (τ 1 , W 1 ), ...(τ i , W i ) sont bien dénis (et nis p.s.). Soit S i = i k=1 τ k . Nous dénissons

U i+1 1 = U W 1 +...+W i +1 , et τ i+1 = inf{t > U i+1 1 , N h ((t -L(h), t]) = 0} -S i , (1.2
W i+1 = N h ([U i+1 1 , S i + τ i+1 ) = N h ([S i , S i + τ i+1 ]), (1.2.16)
et nous renumérotons les sauts de cette fenêtre par :

U i+1 j = U W 1 +...+W i +j , ∀j ∈ {1, ..., W i+1 }.
N h t = ∞ i=1 1 U i ≤t = ∞ i=1 W i j=1 1 U j i ≤t .
(1.2.17)

Introduisons le processus de renouvellement associé aux S i

M h t := ∞ i=1 1 S i ≤t .
(1.2.18)

Puisque S i = i k=1 τ k , on peut dénir

N h t := ∞ i=1 W i 1 S i ≤t = M h t i=1 W i . ( 1
N h t ≤ N h t ≤ N h t + W M h t +1 p.s.
(1.2.20)

On a alors

N h t = M h t i=1 W i + R h t
:= min λ , ||h + || 1 -ln(||h + || 1 )-1 L(h)
on a E(e ατ 1 ) < +∞. ii) Il existe θ 0 > 0 tel que pour tout θ < θ 0 , E(e θW 1 ) < +∞. En particulier τ 1 et W 1 ont des moments polynomiaux de tout ordre. Proposition 1.2.18. Si h ≤ 0, on peut choisir θ 0 < -ln 1 -e -λL(h) dans la proposition précédente 1.2.17.

Grâce à cette première propriété et à la structure de renouvellement des processus de Hawkes, on peut en déduire les résultats asymptotiques suivants : Théorème 1.2.19 (Loi des Grands Nombres). Soit h une fonction signée qui satisfait les Hypothèses 1.2.14 et N h le processsus de Hawkes donné par (1.2.13). Alors

N h t t p.s. -→ t→∞ E[W 1 ] E(τ 1 )
.

λ 1 + λL(h) ≤ E[W 1 ] E(τ 1 ) ≤ λ 1 -||h + || 1 .
Cette méthode nous fournit également un Théorème Central Limite. Alors

√ t N h t t - E[W 1 ] E(τ 1 ) =⇒ t→∞ N (0, σ 2 ) avec σ 2 = Var W 1 -τ 1 E[W 1 ] E(τ 1 ) E(τ 1 )
. Nous dénissons également, pour z ∈ R + ,

J(z) = inf β>0 β Λ * 1 β , z β . De manière similaire, Λ * n et J n sont dénis en remplaçant W 1 par min(W 1 , n) (Ici, les min(W 1 , n) sont déjà à valeurs dans un ensemble ni). Finalement, on dénit J(z) = sup δ>0 lim inf n→∞ inf |y-z|<δ J n (y) .
Grâce à la Proposition 1.2.17 on peut appliquer le Théorème 1.2.7 des processus cumulatifs qui nous indique que le processus approché N h t /t satisfait les inégalités de déviations asymptotiques Théorème 1.2.22. Rappelons que θ 0 est déni dans la Proposition 1.2.17 (ii). Si θ 0 = +∞, la loi de la famille N h t /t satisfait un Principe de Grandes Déviations avec pour fonction de taux J, i.e 1. pour tout fermé C ∈ R,

lim sup t→∞ 1 t ln P N h t /t ∈ C ≤ -inf m∈C J(m) 2. pour tout ouvert O ∈ R, lim inf t→∞ 1 t ln P N h t /t ∈ O ≥ -inf m∈O J(m) , Si θ 0 < +∞, en notant m = E(W 1 )/E(τ 1 ), on a pour tout a > 0 et tout κ ∈ (0, 1) lim sup t→+∞ 1 t ln P N h t t ≥ m + a ≤ -min inf z≥m+κa J(z) , (1 -κ)θ 0 a , et lim sup t→+∞ 1 t ln P N h t t ≤ m -a ≤ -min inf z≥m-κa J(z) , (1 -κ)θ 0 a .
Les dernières inégalités sont obtenues en utilisant le fait que J ≤ J.

Nous prouvons nalement le corollaire suivant Corollaire 1.2.23. On rappelle que θ 0 est déni dans la Proposition 1.2.17 (ii).

(1) Si θ 0 = +∞, N h t /t satisfait le même Principe de Grandes Déviations que

N h t /t.
(2) Si θ 0 < +∞, on a pour tout a > 0 et κ ∈ (0, 1) lim sup

t→∞ 1 t ln P N h t t ≥ m + a ≤ -min inf z-m≥κa J(z) , 1 -κ 2 θ 0 a , (1.2.21)
De façon similaire

lim sup t→∞ 1 t ln P N h t t ≤ m -a ≤ -min inf m-z≤κa J(z) , (1 -κ) 2 θ 0 a . (1.2.22)
avec κ ∈ (0, 1).

Stratégie de preuve

L'idée initiale pour travailler sur les propriétés asymptotiques de ce processus a été d'utiliser la structure sous-jacente de renouvellement montrée par [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF]. C'est ainsi que nous avons pu montrer que les processus de Hawkes, lorsqu'ils sont associés à une fonction h à support compact, pouvaient être considérés comme des processus cumulatifs.

Pour étudier les moments de τ et de W , il susait de rechercher les moments de variables τ + et W + qui apparaissent pour le processus N étendu le travail de [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF] sur des propriétés regénératives de processus de Hawkes. Par ailleurs, la pré-publication [START_REF] Bonde | Renewal Time Points for Hawkes Processes[END_REF] expose également une structure de renouvellement pour des processus de Hawkes dans un cadre général. Il serait donc intéressant de déterminer si ces cadres permettent la comparaison à un processus cumulatif associé.

De plus, l'une des dicultés de la mise en ÷uvre des théorèmes démontrés est l'absence de lien simple entre la fonction de reproduction h et la loi du couple (τ, W ).

Sans forcément déterminer exactement la loi du couple à partir de h, nous pourrions explorer les liens entre des hypothèses sur h et des conséquences sur la loi de (τ, W ). An de modéliser les interactions, nous supposons que tous les neurones sont connectés les uns aux autres, et que les apports de chacun des neurones sont les mêmes.

dX t = (X t -(X t ) 3 -C t -α)dt dC t = (γX t -C t + β)
Ainsi, pour éviter une explosion du modèle, nous supposons les apports de chaque neurone de l'ordre de 1/N : nous travaillons donc dans un cadre champ-moyen.

Ces apports sont décrits par deux fonctions K X et K C , qui sont ajoutés sur la dynamique de chaque coordonnée. Nous nous concentrons sur des interactions chimiques.

Dans ce cas, la fonction d'interaction est appliquée à la diérence entre l'état du neurone qui envoie de l'information et l'état de celui qui le reçoit, i.e. l'apport du neurone j sur le neurone i est donné par

K X (Z i -Z j ) et K C (Z i -Z j ).
Finalement, le modèle de FitzHugh-Nagumo que nous étudions est le suivant : on

considère un réseau de N ∈ N * neurones. Pour 1 ≤ i ≤ N , l'état du i-ème neurone à l'instant t est caractérisé par Z i,N t = (X i,N t , C i,N t ) ∈ R 2 . Les Z i t suivent le système suivant : dX i,N t = (X i,N t -(X i,N t ) 3 -C i,N t -α)dt + 1 N N j=1 K X (Z i,N t -Z j,N t ) + σ X dB i,X t dC i,N t = (γX i,N t -C i,N t + β)dt + 1 N N j=1 K C (Z i,N t -Z j,N t ) + σ C dB i,C t , (1.2.23)
A i xé, Z i,N représente l'état du i-ème neurone dans un réseau de N neurones. On suppose (B i,X t ) t et (B i,C t ) t des mouvements browniens indépendants.

La propagation du chaos

Pour décrire le comportement de ce réseau quand le nombre de neurones tend vers l'inni, une notion importante est la propagation du chaos, que nous redénissons rapidement ici.

Nous travaillons dans le contexte d'une loi symétrique, c'est-à-dire que les particules sont interchangeables.

Dénition 1.2.24. Un N -uplet de variables aléatoires Z N = (Z 1,N , . . . , Z N,N ) suit une loi dite symétrique si cette loi est invariante par permutation du N -uplet. En d'autres termes, pour toute permutation σ : {1, . . . , N } → {1, . . . , N }, la loi de (Z σ(1),N , . . . , Z σ(N ),N ) est la même que la loi de Z N .

En 1956, Kac [START_REF] Kac | Foundations of Kinetic Theory[END_REF] dénit la notion de chaos, en l'appelant "propriété de Boltzmann" -en référence aux travaux de ce dernier sur les collisions d'atomes dans les gazs parfaits. Cette notion a été ensuite développée par Sznitman [START_REF] Sznitman | Topics in Propagation of Chaos[END_REF]. Nous la donnons ici dans R d , mais cette dénition peut se généraliser.

Dénition 1.2.25. Pour tout N ∈ N, soit Z N = (Z 1,N , . . . , Z N,N ) une suite de variables aléatoires de loi symétrique sur (R d ) N . On note ν N la loi de Z N . Soit ν une mesure de probabilité sur R d .

On dit que la suite (ν N ) N est ν-chaotique si pour tout entier k, et pour toutes fonctions tests continues bornées ϕ 1 , . . . , ϕ k , on a

ϕ 1 (x 1 ) × • • • × ϕ k (x k )ν N (dx 1 , . . . , dx k ) -→ N →∞ ϕ 1 (x)ν(dx) × • • • × ϕ k (x)ν(dx) .
Cela signie qu'à k xé, k particules dans un ensemble de N particules sont asymptotiquement i.i.d. quand N tend vers l'inni.

La notion de propagation du chaos en découle assez naturellement : si on considère un système qui part de conditions initiales chaotiques, on souhaite vérier que cette propriété "se propage" au cours du temps. Dénition 1.2.26. Soit un système de particules qui évolue selon une certaine évolution. On suppose que la condition initiale de ce système de N particules suit la loi ν 0,N . On note ν t,N la loi de ce système au temps t. On dit qu'il y a propagation du chaos, si lorsque ν 0,N est chaotique, alors pour tout t > 0, ν t,N est chaotique.

Cela signie qu'à t xé, si le système de N particules est initialement chaotique, alors il le reste à l'instant t. Ainsi, à k et t xés, les k particules parmi N , observées à l'instant t, sont asymptotiquement i.i.d. lorsque le nombre de particules N tend vers l'inni.

Principaux résultats

Pour décrire le comportement du modèle de FitzHugh-Nagumo déni en (1.2.23), nous le comparons au modèle suivant : on dénit le processus ( Zt ) t≥0 = ( Xt , Ct ) t≥0 comme un processus à valeurs dans R 2 qui évolue suivant l'équation diérentielle stochastique de McKean-Vlasov non linéaire suivante

d Xt = ( Xt -( Xt ) 3 -Ct -α)dt + K X * μt ( Zt )dt + σ x d BX t d Ct = (γ Xt -Ct + β)dt + K C * μt ( Zt )dt + σ c d BC t , (1.2.24)
où μt est la loi du processus Zt au temps t et où * décrit l'opération de convolution

K X * μt (u) = K X (u -v)µ t (dv
∃L X ∈ [0, L X,max ), ∀z, z ∈ R 2 |K X (z) -K X (z )| ≤ L X z -z 1 ∃L C ∈ [0, L C,max ), ∀z, z ∈ R 2 |K C (z) -K C (z )| ≤ L C z -z 1 K X (0, 0) = 0 and K C (0, 0) = 0
Pour la propagation du chaos non uniforme en temps, cette hypothèse sur L X,max et L C,max peut être ignorée : dans ce cas, les valeurs choisies peuvent être +∞, et l'important est que les deux fonctions d'interaction K X et K C soient Lipschitz.

Par contre, pour la propagation du chaos uniforme en temps, il sera nécessaire d'avoir des constantes de Lipschitz L X et L C assez petites. Cette condition sur L X,max et L C,max ne sera en fait pas la condition restrictive. Néanmoins, celle-ci permet de dénir d'autres paramètres de manière indépendante de L X et de L C , paramètres qui permettent ensuite de dénir des conditions sur L X et L C . En fait, la Remarque 4.2.7 de ce chapitre indique qu'on peut choisir pour ce théorème L X,max = 4 et L C,max = 1 5 .

On note W 1 et W 2 les distances de Wassertein L

sur R 2 satisfaisant E µ 0 e ã(|X|+|C|) ≤ C0 , on a W 1 µ k,N t , μ⊗k t ≤ B 1 k √ N , W 2 2 µ k,N t , μ⊗k t ≤ B 2 k √ N , pour tout k ∈ N, où µ k,
g K (x) =    -K 3 if x < -K x 3 if x ∈ [-K, K] K 3 if x > K.
Les théorèmes usuelles peuvent alors s'appliquer, et à l'aide d'une fonction de Lyapunov H, quadratique, nous prouvons que les solutions de nos systèmes n'explosent pas.

La seconde partie de cette Proposition, c'est-à-dire l'existence et l'unicité d'une solution de (1.2.24), repose sur un théorème de [START_REF] Gonçalo Dos Reis | FreidlinWentzell LDP in Path Space for McKeanVlasov Equations and the Functional Iterated Logarithm Law[END_REF] dont nous vérions les hypothèses.

Pour démontrer la convergence de µ k,N t vers μ⊗k t quand N tend vers l'inni, nous suivons la méthode de couplage décrite dans [START_REF] Guillin | Convergence Rates for the Vlasov-Fokker-Planck Equation and Uniform in Time Propagation of Chaos in Non Convex Cases[END_REF]. Le résultat de cet article ne peut s'appliquer directement en raison de la dynamique cubique de notre processus.

Cette méthode a été suggérée par Eberle [START_REF] Eberle | Reection Couplings and Contraction Rates for Diusions[END_REF], à la suite de travaux de Lindvall et Rogers [START_REF] Lindvall | Coupling of Multidimensional Diffusions by Reection[END_REF], et a été approfondie par le travail de [START_REF] Durmus | An Elementary Approach to Uniform in Time Propagation of Chaos[END_REF].

Une distance naturelle entre deux mesures de probabilité est la distance de Wasserstein, liée à la théorie de transport optimale. Nous rappelons qu'elle se dénit de la manière suivante : pour µ et ν deux mesures de probabilité sur R d , on note Finalement, nous pouvons dénir une fonction de Lyapunov H pour prendre en compte la tendance de chaque processus à revenir dans un ensemble compact de R 2 .

W p (µ, ν) = inf X∼µ, Y ∼ν E ||X -Y || p p 1/p , (
Nous étudions ensuite une distance adaptée entre les processus, qui dépend de la fonction de Lyapunov an de prendre en compte la structure du système de particules. Cette distance est de la forme ρ

t := 1 N N i=1 f (r i t )(1+ H( Zi t )+ H(Z i,N t ))
, avec > 0. Cette quantité permet de contrôler les distances L 1 et L 2 usuelles. Lorsque r i t est petit, f (r i t ) va décroître, soit grâce à la partie déterministe, soit grâce au couplage par réexion. Lorsque r i t est grand, c'est la partie avec la fonction de Lyapunov H qui va décroître. Cela nous permet de montrer que E(ρ t ) décroît exponentiellement vite. Bien sûr, cela mène à un certain nombre de contraintes sur les diérents paramètres et nous vérions que ces contraintes peuvent bien être satisfaites. La distance ρ t sera légèrement modiée pour prendre en compte la non-linéarité du processus.

La méthode décrite ci-dessus a bien sûr besoin de bruit dans la direction orthogonale au sous-espace naturellement contractant. Ainsi, il est nécessaire d'avoir σ X > 0. Dans le cas où σ X = 0 et σ C > 0, un autre sous-espace est considéré et les calculs sont en conséquence modiés. Ce cas est traité en annexe. Comme cela a été présenté rapidement dans l'introduction, de nombreuses questions sont soulevées pour les modèles neuronaux, aussi ce travail a de nombreuses directions d'extensions. Tout d'abord, certains auteurs [START_REF] Salem | A Gradient Flow Approach of Propagation of Chaos[END_REF] se sont intéressés à la notion de bruit dit environnemental dans d'autres cadres. Il s'agit d'un bruit identique pour chacune des particules : l'idée est d'avoir B i,X = B X et B i,C = B C . Nous pourrions nous intéresser au comportement à long-terme de ce type de processus, ou de processus présentant à la fois un bruit environnemental et un bruit individuel.

Par ailleurs, [START_REF] Luçon | Periodicity Induced by Noise and Interaction in the Kinetic Mean-Field FitzHugh-Nagumo Model[END_REF] et [START_REF] Bossy | Synchronization of Stochastic Mean Field Networks of Hodgkin-Huxley Neurons with Noisy Channels[END_REF] s'est concentré sur des propriétés de synchronisation neuronales : il a été observé, biologiquement, que certains neurones se synchronisent et se déchargent de manière groupée, les décharges ayant alors lieu sur des plages de temps assez réduites. Certains modèles permettent de modéliser cette synchronisation, en particulier le modèle de FitzHugh-Nagumo, dans un cadre dans lequel la dynamique du drift est "ralenti" devant la dynamique générée par l'interaction et le bruit.

Nous pourrions également étendre notre étude en considérant des paramètres non plus globaux mais individuels. On peut envisager des paramètres α i , β i et γ i pour chacun des neurones i, qui suivent une certaine répartition autour de paramètres moyens α, β et γ. Les mêmes questions que ci-dessus pourraient alors être posées : la synchronisation étudiée dans [START_REF] Luçon | Periodicity Induced by Noise and Interaction in the Kinetic Mean-Field FitzHugh-Nagumo Model[END_REF] est vraie pour des paramètres appartenant à un certain ensemble. Qu'en serait-il si le jeu de paramètres d'une partie des neurones est dans cet ensemble, et celui du reste des neurones n'y est pas ? Aurait-on un entraînement généré par les neurones synchronisés qui se propagerait aux autres ? Si c'est le cas, à partir de quelle proportion de neurones dans le "bon" ensemble de paramètres ? L'article [4], complété par [START_REF] Bossy | Clarication and Complement to Mean-Field Description and Propagation of Chaos in Networks of HodgkinHuxley and FitzHughNagumo Neurons[END_REF], considère des interactions plus élaborées pour les processus de FitzHugh-Nagumo. Ici, comme beaucoup dans la littérature, nous nous sommes focalisés sur des synapses dites électriques, qui peuvent être représentées comme une fonction (K X ou K C ) de la diérence entre deux états Z i -Z j . Les synapses chimiques nécessitent une autre équation dans le système d'EDS, ce qui augmente la complexité du modèle.

Nous voudrions également généraliser notre étude en ne nous limitant pas à des interactions Lipschitz, et en considérant par exemple des fonctions discontinues qui peuvent représenter des eets seuils, ou des interactions ne prenant pas en compte l'état du neurone i étudié (qui ne dépendraient que de l'état des Z j ).

Enn, si ici nous nous sommes concentrés sur le modèle FitzHugh-Nagumo à 2 EDS par neurone, nous pourrions explorer le modèle Hodgkin-Huxley à 4 EDS par neurone (voire plus si des synapses chimiques sont prises en compte) et vérier si les outils mis en place ici peuvent s'envisager. Par exemple, [4] et [START_REF] Bossy | Clarication and Complement to Mean-Field Description and Propagation of Chaos in Networks of HodgkinHuxley and FitzHughNagumo Neurons[END_REF]

Cumulative processes

Cumulative processes have been introduced by Smith [START_REF] Walter | Regenerative Stochastic Processes[END_REF] and are applied in many purposes, such as nance where they are called compound-renewal processes or renewal-reward processes. Indeed these continuous time processes cumulate independent random variables occurring in time interval given by a renewal process.

To be more specic a real valued process (Z t ) t≥0 is called a cumulative process if the following properties are satised:

1. Z 0 = 0, 2. there exists a renewal process (S n ) n such that for any n, (Z Sn+t -Z Sn ) t≥0 is independent of S 0 , ...S n and (Z s ) s<Sn , 3. the distribution of (Z Sn+t -Z Sn ) t≥0 is independent of n.

To study such processes, we write for all t ≥ 0

Z t = W 0 (t) + W 1 + ... + W Mt + r t where W 0 (t) = Z t∧S 0 , W n = Z Sn -Z S n-1 , r t = Z t -Z Mt , where M t is dened by M t = sup {n ≥ 0, S n ≤ t} .
The (W k ) k≥1 's are thus i.i.d.

We denote by (τ i ) i the times associated to the renewal process τ n = S n -S n-1 and τ 0 = 0.

It is worth noticing that τ i and W i can be dependent.

In the sequel we suppress the subscript i when dealing with the distribution (and all associated quantities like expectation, variance ...) of (τ i , W i ) and simply use (τ, W ).

A simple example of cumulative process is Z t = t 0 f (X s )ds where (X t ) t is a regenerative process with i.i.d. cycles [START_REF] Glynn | Limit Theorems for Cumulative Processes[END_REF]. Markov additive processes are other classical examples of cumulative process.

In [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF] the authors exhibited a renewal structure for some Hawkes processes. This description is extensively used in our companion paper [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF] in order to describe such processes as cumulative processes, and to study their asymptotic behaviour.

For cumulative processes, the law of large numbers (assuming that

E[|W |] and E[τ ] are not innite) Z t t a.s. -→ t→∞ E[W ] E[τ ] if and only if E max S 0 ≤t<S 1 |r t | < ∞ ,
and the central limit theorem (assuming Var(W ) < ∞ and Var(τ ) < ∞)

Z t -t E[W ] E[τ ] √ t -→ t→∞ N 0, σ 2 where σ 2 = 1 E(τ ) Var W - E[W ] E[τ ]
τ can be found in Asmussen [START_REF] Asmussen | Applied Probability and Queues. 2nd ed. Stochastic Modelling and Applied Probability[END_REF], theorem 3.1 and theorem 3.2.

Brown and Ross [START_REF] Brown | Asymptotic Properties of Cumulative Processes[END_REF] have proved equivalent of the Blackwell theorem and of the key renewal theorem for a subclass of cumulative processes, since cumulative processes are a generalization of renewal processes. Glynn and Whitt have focused in [START_REF] Glynn | Limit Theorems for Cumulative Processes[END_REF] on cumulative processes associated to a regenerative process and have proved law of large numbers (strong and weak), law of the iterated logarithm, central limit theorem and functional generalizations of these properties.

The aim of this work is to establish a large deviation principle (LDP) for cumulative processes. Some works have already been done. Duy and Metcalfe [START_REF] Duy | How to Estimate the Rate Function of a Cumulative Process[END_REF] have considered the estimation of a rate function for a cumulative process (if it admits a LDP). In a series of papers, Borovkov and Mogulskii ([9,[START_REF] Borovkov | Large Deviation Principles for Trajectories of Compound Renewal Processes[END_REF][START_REF] Borovkov | Large Deviation Principles for Trajectories of Compound Renewal Processes[END_REF]) have studied the (LDP) (they use the term compound-renewal theorem), under some assumptions of comparison between the values (θ 1 , θ 2 ) for which E e θ 1 τ +θ 2 W is nite and the value θ for which E e θτ is nite (a comparison between the joint distribution of (τ, W ) and the law of τ ). Actually, some points in their approach are not clear for us.

Lefevere, Mariani and Zambotti [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] worked on specic cumulative process where W i = F (τ i ) for some deterministic function F which is assumed to be non-negative, bounded and continuous. We choose to extend their approach. Our proof of the LDP has the same skeleton, but in a general framework for the pair (τ, W ).

Chapter 2. LDP for cumulative processes

In this paper, we show a LDP for Z t /t in the case r t = 0. This assumption can be relaxed if r t /t tends to 0 quickly enough, as it will be the case for the application to Hawkes process (see [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF]). For example, if for all δ > 0 lim sup

t→∞ 1 t ln P |r t | t > δ = -∞, then Z t /

Hawkes processes

A Hawkes process is a point process on the real line R characterized by its intensity process t → Λ(t). We consider an appropriate ltered probability space (Ω, F, (F t ) t≥0 , P) satisfying the usual assumptions. Denition 2.1.1. Let λ > 0 and h : (0, +∞) → R a signed measurable function.

Let N 0 a locally nite point process on (-∞, 0] with law m. The point process N h on R is a Hawkes process on (0, +∞), with initial condition N 0 and reproduction measure µ(dt) = h(t)dt if:

N h | (-∞,0] = N 0 , the conditional intensity measure of N h | (0,+∞) with respect to (F t ) t≥0 is absolutely continuous w.r.t the Lebesgue measure and has density:

Λ h : t ∈ (0, +∞) → f λ + (-∞,t)
h(t -u)N h (du) .

(2.1.1)

for some non-negative function f .

Hawkes processes have been introduced by Hawkes [START_REF] Hawkes | Spectra of Some Self-Exciting and Mutually Exciting Point Processes[END_REF]. Most of the literature concerned with the large time behaviour of N h t = N h ([0, t]) is dedicated to the case h ≥ 0 (self excitation). This behaviour is studied in details in [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF] when h is a signed (the negative part modelling self inhibition) compactly supported function, and the function f (called the jump rate function) is given by f (u) = max(0, u) .

In this situation one gets a description of N h t as a cumulative process (see [20] subsection 2.3) with few information on the joint law of (τ, W ). This was the initial motivation for the present work. We refer to [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF] for a more complete overview and explicit results in this situation. In the remaining part of the paper we will come back to more general cumulative processes.

Notations and main result 2.2.1 First notations.

We consider (τ 1 , W 1 ), (τ 2 , W 2 ), ... an i.i.d. sequence of pairs of random variables built on some probability space (Ω, P). The law of (τ i , W i ) is an arbitrary probability ψ on (0, +∞) × R. We denote this by: (τ i , W i ) ∼ ψ. In the sequel we generically use the notation (τ, W ) for a pair with the same distribution as (τ i , W i ). Notice that we assume in particular that E(τ ) > 0.

We denote by M 1 (X ) the space of probability measure on some measurable space X .

We consider the renewal process associated with (τ i ) i≥1 :

S 0 = 0, S n = n k=1 τ k , M t = sup {n ≥ 0, S n ≤ t} .
We will study the quantity:

Z t = Mt i=1 W i . (2.2.1)
The main goal of this paper is to prove a Large Deviation Principle for the process Z t /t. Let us recall some basic denitions in large deviation theory (we refer to [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]).

A family of probability measures (η t ) t∈R + on a topological space (X , T X ) satises the Large Deviations Principle (LDP) with rate function J(.) and speed γ(t) = t if J is lower semi-continuous from X to [0, +∞], and the following holds In this paper we only consider the speed function γ(t) = t so that we will no more refer to it.

-inf x∈O J(x) ≤ lim
A particularly important notion for our purpose is the notion of exponentially good approximation.

i) ∃θ 0 ∈ (0, +∞] such that E[e θτ ] < ∞ for θ < θ 0 , ii) ∃β 0 ∈ (0, +∞] such that E[e β|W | ] < ∞, for β < β 0 .
We will introduce the classical Cramer transforms, for (a, b) ∈ R We may replace a/2 and β 0 /2 by κa and (1 -κ)β 0 for any κ ∈ (0, 1).

J n (m) = inf β>0 β Λ * n 1 β , m β , J(m) = inf β>0 β Λ * 1 β , m β , ( 2
Remark 2.2.6. The previous inequalities (2.2.8) and (2.2.9) actually hold true with function J as well as function J since J ≤ J (see Lemma 2.9.5). However, since J is clearly more easy to calculate than J, we prefer to write the inequalities with J.

2.2.3

The scheme of proof.

The proof will be divided in several steps. In the next section we will reduce the problem rst to bounded W 's replacing the original W by W n , and then to nite valued W 's. This will be done by using exponentially good approximations and Theorem 2.2.2.

Following [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] we introduce in section 2. At this point the fact that W is nite valued will provide us both with simple conditioning and with the necessary compactness arguments, since M 1 ({1, ..., n}) is compact.

If many arguments are close to those in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF], some have to be written in detail.

We have decided to refer to the corresponding statements in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] only when they can be reproduced line by line in our case. Some topological points also have to be claried. In section 2.6 we show useful auxiliary lemmata. The next section 2.7 is devoted to the proof of 2.4.3. In section 2.8 we deduce theorem 2.4.4. The nal section 2.9 is devoted to the study of the rate function J and to the proof of theorem 2.2.5.

2.3

Reduction of the problem to nite valued W 's.

First reduction to bounded W 's.

To Z t dened by (2.2.1) we associate

Z n t = Mt i=1 (W i ∧ n ∨ (-n)) , so that |Z t -Z n t | = Mt i=1 (W i -n) + + Mt i=1 (W i + n) - (2.3.1)
where u + = max(u, 0) and u -= max(-u, 0). We then have Lemma Using that the W i 's are i.i.d. we may write for δ > 0 and c > 0, (as usual an empty sum is equal to 0 by convention)

P Mt i=1 (W i -n) + > δt ≤ P ct i=1 (W i -n) + > δt 2 + P Mt i=ct+1 (W i -n) + > δt 2 ≤ P ct i=1 (W i -n) + > δt 2 + P Mt i=ct+1 (W i -n) + > δt 2 ∩ {1 + ct ≤ M t < 2ct} + P Mt i=ct+1 (W i -n) + > δt 2 ∩ {M t ≥ 2ct} ≤ 2P   ct j=1 (W j -n) + > δt 2   + P (M t ≥ 2ct)
Study of P (M t ≥ 2ct) . Start with the second term in the sum above.

According to theorem 2.3 in [START_REF] Tiefeng | Large Deviations for Renewal Processes[END_REF], we know that M t /t satises a LDP with rate function J τ given by

J τ (x) = sup λ {λ -x ln E(e λτ )} if x ≥ 0 ∞ if x < 0
Notice that J τ (x) = x Λ * (1/x, 0) for x > 0. In addition (see Lemma 2.6 in [START_REF] Tiefeng | Large Deviations for Renewal Processes[END_REF]) the supremum is achieved for λ ≤ 0 if x ∈ (1/E(τ ) , +∞) and J τ is non-decreasing on this interval.

It follows that for 2c > 1/E(τ ), lim sup

t→∞ 1 t ln P (M t ≥ 2ct) = -J τ (2c) . (2.3.3)
In order to get lim n lim sup t→+∞ 1 t ln P (M t ≥ 2c n t) = -∞ for some sequence c n (to be chosen later) it remains to show that

J τ (x) -→ x→∞ +∞.
Since τ is a non-negative random variable one can nd λ τ such that E(e λτ τ ) = e -1 .

Let x ∈ R + , we have:

J τ (x) = sup λ {λ -x ln E(e λτ )} ≥ λ τ -x
P   ct j=1 (W j -n) + > δt 2   = P   ct j=1 (W j -n) + > δt 2
  so that we may use this time the usual Cramer's theorem. Dening

Ψ n (λ) = ln E e λ(W -n) + Ψ * n (x) = sup λ {λx -Ψ n (λ)},
we have lim sup

t→∞ 1 t ln P   ct j=1 (W j -n) + > δt/2   = lim sup t→∞ c ct ln P   ct j=1 (W j -n) + > δt/2   ≤ lim sup t→∞ c ct ln P   ct j=1 (W j -n) + > δ ct /2c   ≤ -c inf x∈[δ/2c,+∞) Ψ * n (x). As the function x → Ψ * n (x) is non-decreasing on [E((W -n) + ), +∞), we have lim sup n→∞ 1 t ln P   ct j=1 (W j -n) + > δt/2   = -c Ψ * n (δ/2c) , provided δ/2c ≥ E((W -n) + ). Notice that for λ < β 0 , c Ψ * n (δ/2c) ≥ λδ 2 -c ln 1 + E (e λ(W -n) -1) 1 W >n
so that choosing c n growing to innity and such that

c n E((W -n) + ) → 0 and c n ln 1 + E (e λ(W -n) -1)1 W >n → 0 as n → ∞ which is always possible since both E((W -n) + ) and ln 1 + E (e λ(W -n) -1)1 W >n are going to 0, we get lim n lim sup t→∞ 1 t ln P   ct j=1 (W j -n) + > δt/2   ≥ λδ 2 .
We may optimize in λ and plugging the same sequence c n in (2.3.3) ends the proof. Starting with a bounded W such that -K < W < K almost surely we dene a discretized version of W by

W n = n-1 j=-n jK n 1 W ∈[jK/n,(j+1)K/n[ . It clearly holds |W -W n | ≤ K n .
We thus introduce

Zn t = Mt i=1 W n i so that | Zn t -Z t | ≤ K M t n .
According to the study in the previous section lim sup

t→∞ 1 t ln P Zn t t - Z t t > δ ≤ lim sup t→∞ 1 t ln P (M t ≥ δtn/K) = -J τ (δn/K)
W n = -n1 W <-n + n1 W ≥n + n 2 -1 j=-n 2 j n 1 W ∈[ j n , j+1 n ) . By denoting Zn t = Mt i=1 W n i , we have, for δ > 0 P Z t t - Zn t t > δ ≤P Mt i=1 (W i -n) + > δt + P Mt i=1 (W i + n) + > δt + P   Mt i=1 n 2 -1 j=-n 2 W i - j n 1 W i ∈[ j n , j+1 n ) > δt   . Since Mt i=1 n 2 -1 j=-n 2 W i - j n 1 W i ∈[ j n , j+1 n ) ≤ M t n , we obtain P   Mt i=1 n 2 -1 j=-n 2 W i - j n 1 W i ∈[ j n , j+1 n ) > δt   ≤ P(M t > δtn).
Zn t t > δ ≤ - β 0 δ 2 .
In particular if β 0 = +∞, Zn t /t is an exponentially good approximation of Z t /t.

Remark 2.3.5. The proofs in the previous section suggest a direct naive approach in order to get deviation bounds. Indeed we may write

Z t t - E(W ) E(τ ) ≤ 1 t Mt i=1 (W i -E(W )) + E(W ) M t t - 1
E(τ )
so that following the lines of the proof of lemma 2.3.1, we may write

P Z t t - E(W ) E(τ ) ≥ δ ≤ P   1 t Mt i=1 (W i -E(W )) ≥ δ/2   + P E(W ) M t t - 1 E(τ ) ≥ δ/2 ≤ 2P ct i=1 (W i -E(W )) > δt/4 + P( M t > 2ct) + + P E(W ) M t t - 1
E(τ ) ≥ δ/2 .
Introducing the Cramer transform of W ,

Ψ * (x) = sup λ {λx -ln E(e λ(W -E(W )) )} we thus deduce, if Assumption 2.2.4 is fullled lim sup t→∞ 1 t ln P Z t t - E(W ) E(τ ) ≥ δ ≤ -A(δ) , (2.3.4)
where

A(δ) = min(A 1 (δ), A 2 (δ)) A 1 (δ) = min J τ 1 E(τ ) + δ 2E(W ) , J τ 1 E(τ ) - δ 2E(W ) (2.3.5) A 2 (δ) = sup 2c>1/E(τ ) min (J τ (2c) , c min (ψ * (δ/4c), ψ * (-δ/4c))) . (2.3.6)
This time however we cannot let c go to innity so that A 2 (δ) will furnish a worse bound than β 0 δ/2 (the correcting term in Lemma 2.3.1). So even when β 0 < +∞ the previous bound will be worse than the one obtained by combining Lemma 2.3.1

and the LDP for nite valued W .

This kind of approach should be useful in order to get non asymptotic bounds. ♦ 2.4 Large deviations for a nite valued W .

From now on we will assume that W takes its values in a set of cardinal n (xed). However we will keep a superscript n to remember that this is a reduction of the general case, so we shall write W n . For simplicity we rename the values of W n as 1, ..., n. According to Assumption 2.2.4 iii) and the approximation made in subsection 2.3.2 we may assume that ∀j = 1, ..., n , ∀t > 0 , ψ n ({τ > t} ∩ {w = j}) = P(τ > t, W n = j) > 0 .

This condition will replace iii) in Assumption 2.2.4 in all the next statements. Some results are still true when W takes its value in some W ⊂ R which is compact, in particular if W is bounded. We will mention these results in related remarks each time it is possible.

More notations.

As we said we will now closely follow the approach in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. Consider the backward recurrence time process (A t ) t≥0 , the forward recurrence time process (B t ) t≥0 and the process (C n t ) t≥0 , dened by:

A t = t -S Mt , B t = S Mt+1 -t, C n t = W n Mt+1 .
Note that for all time t the random variable A t + B t = S Mt+1 -S Mt is distributed as τ . In the sequel, we study the associated empirical measure:

µ n t := 1 t [0,t) δ (As,Bs,C n s ) ds ∈ M 1 ((0, +∞] 2 × {1, ...,

n}).

We denote by P n t the law of µ n t .

This empirical measure is a probability measure on (0, +∞] 2 × {1, ..., n}. Unlike in the work of Lefevere, Mariani and Zambotti [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] who focused on the specic case where W = F (τ ), we have to consider three coordinates in the empirical measure.

Looking at (0, +∞] allows us to avoid integrability considerations at innity since A s and B s can be as large as we want simultaneously. Choosing as metric d(t, t ) = | 1 t -1 t | on (0, +∞] and the usual one on {1, ..., n} we immediately see that Y = (0, +∞] 2 ×{1, ..., n} is a Polish space, so that M 1 ((0, +∞] 2 × {1, ..., n}) is also Polish. In addition (a, b, c) → 1 a+b is continuous (of course it is equal to 0 if either a or b equals +∞).

For µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), we denote:

µ(1/τ ) = µ(1/(a + b)) = (0,+∞) 2 ×{1,...,n} 1 a + b µ(da, db, dc)
and for π ∈ M 1 ((0, +∞] × {1, ..., n}), we denote:

π(1/τ ) = (0,+∞]×{1,...,n} 1 τ π(dτ, dW ).
Let us dene ∆ n 0 ⊂ M 1 ((0, +∞] 2 × {1, ..., n}) as the set of measures such that a and b correspond to a uniform partition of the length τ , i.e. a = uτ and b = (1 -u)τ where u is a uniform random variable independent of (τ, W ):

∆ n 0 = {µ 0 ∈ M 1 ((0, +∞] 2 × {1, ..., n}), (2.4.1)
µ 0 (da, db, dc) = [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) (da, db, dc)du ⊗ π(dτ, dW ), π ∈ M 1 ((0, +∞) × {1, ..., n}), π(1/τ ) < +∞}
We also dene ∆ n contains mixing of an element of ∆ n 0 and a measure on (+∞, +∞)× {1, . . . , n} described by the law η:

∆ n = {µ(da, db, dc) = αµ 0 (da, db, dc) + (1 -α)δ (+∞,+∞) (da, db) ⊗ η(dc), µ 0 ∈ ∆ n 0 , α ∈ [0, 1], η ∈ M 1 ({1, ..., n})}. (2.4.2)
The following will be used several times in what follows Lemma 2.4.1. For M > 0, the set

X M = {ν ∈ M 1 ((0, +∞] 2 × {1, ..., n}) , ν(1/(a + b)) ≤ M } is compact in M 1 ((0, +∞] 2 × {1, ..., n}). Proof. First the set {a + b ≤ δ} is compact in (0, +∞] 2 × {1, ..., n} for δ ≥ 0. It follows that X M is a uniformly integrable set, hence is relatively compact (tight). Indeed for ν ∈ X M ν({a + b < δ}) ≤ δ ν(1/(a + b)) ≤ δ M .
Notice that this set is also closed, so that it is compact. Indeed Recall that ψ n denotes the joint distribution of (τ, W n ), so that its rst marginal does not depend on n. We may thus skip the superscript n when dealing with quantities that only depend on this rst marginal. To simplify the notation we will denote by ξ the θ 0 in Assumption 2.2.4, i.e.

if ν belongs to X M , ν( 1 a+b+ε ) ≤ M for all ε > 0. Any weak limit ν of such ν k 's satises ν( 1 a+b+ε ) ≤ M since (a, b, c) → 1/(a + b + ε)
ξ = sup{c ∈ R, ψ(e cτ ) < ∞} ∈ [0, +∞]. (2.4.3) For π ∈ M 1 ((0, +∞] × {1, ..., n}), satisfying π(1/τ ) ∈ (0, +∞), we dene π ∈ M 1 ((0, +∞] × {1, ..., n}) as: π(dτ, dW ) = 1 π(1/τ ) 1 τ π(dτ, dW ).
π has no weight on {+∞} × {1, ..., n}.

We also dene the functional I n 0 , I n : M 1 ((0, +∞] 2 × {1, ..., n}) → [0, +∞] by:

I n 0 (µ) = π(1/τ )H(π|ψ n ) if µ ∈ ∆ n 0 ∞ otherwise , (2.4.4)
I n (µ) = απ(1/τ )H(π|ψ n ) + (1 -α)ξ if µ ∈ ∆ n ∞ otherwise , (2.4.5)
where H is the relative entropy dened by

H(ν|µ) =      ln dν dµ dµ if ν is absolutely continuous w.r.t. µ ∞ otherwise.
We denote by C b ((0, +∞] 2 × {1, ..., n}) the set of bounded and continuous functions on (0, +∞] 2 × {1, ..., n}. We introduce a subspace Γ, which will be useful to get a control on the functional I n .

For a bounded measurable f : (0, +∞) × (0, +∞) × {1, ..., n} → R, we set :

f (x, τ, W ) = x 0 f (uτ, (1 -u)τ, W )du (2.4.6)
We dene:

C n,f = (0,+∞)×{1,...,n} e τ f (1,τ,W ) ψ n (dτ, dW ) , (2.4.7)
and

D n,f = sup s>0 (s,+∞)
e τ f (s/τ,τ,W ) ψ n (dτ, dW ) ..

Chapter 2. LDP for cumulative processes

We consider

Γ := {f : (0, +∞] 2 × {1, ..., n} → R, bounded, lower semicontinuous, C n,f < 1, D n,f < +∞}.
(2.4.9)

The reasons for introducing such quantities are detailed in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] section 1, see in particular subsection 1.5 (and more particularly Remark 1.5) and subsection 1.6.

2.4.2 Large deviations principle for cumulative process when W is nite valued.

We will rst prove a LDP for the empirical measure µ n t :

Theorem 2.4.3. Let µ n t the empirical measure of a cumulative process satisfying Asumptions 2.2.4, and such that W n takes its values in {1, ..., n}. The family (P n t ) t≥0 of probability distributions of µ n t satises a large deviations principle with good rate function I n as t → ∞ with speed t, where I n is dened in (2.4.5).

Applying the contraction principle we will then deduce

Theorem 2.4.4. Let (τ i , W n i ) i an i.i.d. sequence of couples of random variables in (0, +∞) × {1, ..., n} following the law ψ n . Let us dene Z n t = Mt i=1 W n i the associated cumulative process. If ψ n satises Asumptions 2.2.4
, then the law of Z n t /t satises a large deviation principle with good rate function J n (given by (2.2.6)).

The functional I n

This section aims at proving that I n , dened in (2.4.5), is a good rate function i.e proposition 2.5.2 (analogue to Proposition 1.3 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. Three propositions are necessary to prove it: the propositions 2.5.3, 2.5.4 and 2.5.6 which are the analogues of Proposition 2.1 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. In order to prove these three propositions some additional lemmata are necessary. Remark 2.5.1. If the scheme of proof is close to the one in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF], the proofs of this section need some new ideas. In particular, we will use conditioning by sets like {W = j}. This induces to use disintegration and uniform controls in j. That is why it is useful to work with a nite valued W . In addition the compactness of M 1 ({1, ..., n}) will help to control the third coordinate (which does not exist in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]). Except two auxiliary lemmata, we decided to give self contained proofs. ♦ Proposition 2.5.2. The function I n is a good rate function. Moreover, I n is the lower semicontinuous envelope of I n 0 .

For the proof we need the next three propositions. We need another notation: if ψ k is a sequence in M 1 ((0, +∞) × {1, ..., n}), ξ k and I n k are dened as in (2.4.3) and (2.4.5) respectively, with ψ n replaced by ψ k . Proposition 2.5.3. Let ψ k be a sequence in M 1 ((0, +∞) × {1, ..., n}). Assume that ψ k ψ n , and ξ k → ξ as k → +∞. Then any sequence (µ k ) k in M 1 ((0, +∞] 2 × {1, ..., n}) such that lim sup k→∞ I n k (µ k ) < ∞ is tight and thus, relatively compact in M 1 ((0, +∞] 2 × {1, ..., n}). In particular, for k large enough, such a sequence is in ∆ n . If we consider π k and α k the associated quantities, we have:

lim sup k→∞ α k π k (1/τ ) < +∞.
(2.5.1) Proposition 2.5.4. Let ψ k be a sequence in M 1 ((0, +∞) × {1, ..., n}). Assume that ψ k ψ n , and ξ k → ξ as k → +∞. Then for any µ and any sequence

(µ k ) k in M 1 ((0, +∞] 2 × {1, ..., n}), such that µ k µ, we have lim inf k→∞ I n k (µ k ) ≥ I n (µ).
Remark 2.5.5. Both previous propositions are still true when replacing {1, ..., n} by W a compact subset of R as it will be clear looking at their proof. ♦ Proposition 2.5.6. Let ψ k be a sequence in M 1 ((0, +∞) × {1, ..., n}). Assume that

ψ k ψ n , and ξ k → ξ as k → +∞. Then for any µ in M 1 ((0, +∞] 2 × {1, ..., n}) with I n (µ) < ∞, there exists a sequence (µ k ) k such that µ k µ, µ k ∈ ∆ n 0 for all k and lim sup k→∞ I n k (µ k ) ≤ I n (µ). Moreover, we have µ k 1 a+b -→ k→∞ µ 1 a+b .
For this last proposition we need to work with W taking values in a nite set. Now we can prove the main Proposition of this section.

Proof of Proposition 2.5.2. We want to prove that I n is a good rate function and is the lower semicontinuous envelope of I n 0 .

We apply Proposition 2.5.3 with ψ k = ψ n : for all (µ k ) k sequence of M 1 ((0, +∞] 2 × {1, ..., n}), if lim sup k→∞ I n (µ k ) < +∞, then the family (µ k ) k is tight in M 1 ((0, +∞] 2 × {1, ..., n}). We can deduce that I n has relatively compact sublevel sets (and is coercive).

Thanks to Proposition 2.5.4 we have, for any µ and any sequence µ k in M 1 ((0, +∞] 2 × {1, ..., n}) such that µ k µ, lim inf k→∞ I n (µ k ) ≥ I n (µ). Thus I n has closed sublevel sets. It also means that I n is lower semicontinuous. I n has relatively compact and closed sublevel sets, so I n is a good rate function.

We already know that I n ≤ I n 0 since ∆ n 0 ⊂ ∆ n , and I n is lower semicontinuous, so I n is smaller than or equal to the lower semicontinuous envelope of I n 0 .

Thanks to proposition 2.5.6, for any µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}) with I n (µ) < ∞, there exists (µ k ) k a sequence in ∆ n 0 such that µ k µ and lim sup k→∞ I n (µ k ) ≤ I n (µ). In fact, lim sup k→∞ I n 0 (µ k ) ≤ I n (µ). So I n is greater or equal to the lower semicontinuous envelope of I n 0 and nally I n is the lower semicontinuous envelope of I n 0 .

We turn to the proof of the three auxiliary propositions.

Proof of proposition 2.5.3. Choose a sequence (µ k ) k in M 1 ((0, +∞] 2 × {1, ..., n}) such that lim sup k→∞ I n k (µ k ) < ∞. We want to prove that this sequence is tight and thus, relatively compact in M 1 ((0, +∞] 2 × {1, ..., n}).

Since lim sup k→∞

I n k (µ k ) < ∞, for k large enough, µ k ∈ ∆ n . Then, there exist α k ∈ [0, 1], π k ∈ M 1 ((0, +∞) × {1, ..., n}) with π k (1/τ ) < +∞ and η k ∈ M 1 ({1, ..., n}).
First, we prove equation (2.5.1):

lim sup k→∞ α k π k (1/τ ) < +∞. We have I n k (µ k ) = α k π k (1/τ )H(π k |ψ k )+(1-α)ξ k ≥ α k π k (1/τ )H(π k |ψ k ). Therefore α k π k (1/τ ) ≤ I n k (µ k ) H(π k |ψ k ) (we consider H(π k |ψ k ) = 0). Moreover α k π k (1/τ ) = α k 1 πk (τ )
.

Then

α k π k (1/τ ) ≤ I n k (µ k ) H(π k |ψ k ) ∧ α k πk (τ ) and lim sup k→∞ α k π k (1/τ ) ≤ lim sup k→∞ I n k (µ k ) H(π k |ψ k ) ∧ α k πk (τ ) . α k ∈ [0, 1] and lim sup k→∞ I n k (µ k ) < +∞, so there exists a constant C < ∞ such that ∀k, α k ≤ C and I n k (µ k ) ≤ C. Then lim sup k→∞ α k π k (1/τ ) ≤ lim sup k→∞ C H(π k |ψ k ) ∧ C πk (τ ) ≤ C lim sup k→∞ 1 H(π k |ψ k ) ∧ 1 πk (τ ) ≤ C lim sup k→∞ 1 H(π k |ψ k ) ∨ πk (τ ) ≤ C lim inf k→∞ H(π k |ψ k ) ∨ πk (τ )
.

If there exists a subsequence k j such that H(π k j |ψ k j ) → 0, then lim πk j = lim ψ k j = ψ n . Thus, since [0, 1] and M 1 ({1, ..., n}) are compact, there exist subsequences of α k and η k which converge. Eventually, there exists a subsequence of µ k which converges, so this sequence is relatively compact. In particular, (2.5.1) is true because lim inf j→∞ πk j (τ ) = ψ n (τ ) > 0.

Else, lim inf k→∞ H(π k |ψ k ) > 0 and lim sup k→∞ α k π k (1/τ ) < ∞. Then (2.5.1) is true. Moreover, for M large enough and k large enough, µ k (1/(a + b)) = α k π k (1/τ ) ≤ M . So µ k ∈ X M which is a compact set by Lemma 2.4.1.
Then, this sequence is relatively compact. In both cases, the sequence is relatively compact and (2.5.1) is true.

Remark 2.5.7. As already said we only used the compactness of M 1 (W) and lemma 2.4.1 so that the above proof immediately extends to W compact. ♦

Proof of proposition 2.5.4. Let µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}) and let

(µ k ) k in M 1 ((0, +∞] 2 × {1, ..., n}), such that µ k µ. We want to prove that lim inf k→∞ I n k (µ k ) ≥ I n (µ).
Since µ k µ, we may replace µ k by subsequences again denoted µ k in what follows. We assume

sup k I n k (µ k ) < ∞ (otherwise, if lim inf k→∞ I n k (µ k ) = +∞, +∞ ≥ I n (µ)). Then µ k ∈ ∆ n and we denote by α k , π k , η k and µ 0,k the correspond- ing quantities, with α k ∈ [0, 1] and π k (1/τ ) < ∞.
First, if lim sup k→∞ α k = 0, we may assume taking a subsequence that lim

α k = 0. So µ = δ (+∞,+∞) ⊗ (lim k η k ). Since M 1 ({1, ..., n}) is compact, (lim k η k ) ∈ M 1 ({1, ..., n}). Therefore lim inf k→∞ I n k (µ k ) = lim inf k→∞ α k π k (1/τ )H(π k |ψ k )+(1-α k )ξ k ≥ lim inf k→∞ (1-α k )ξ k = ξ = I n (µ).
Secondly, we have lim sup k→∞ α k = ᾱ > 0. Again we may assume that lim α k = ᾱ.

We begin by studying the sequence µ k and π k to have some information about lim inf k→∞ I n k (µ k ). Then, we prove that µ is in ∆ n , and nally we work on I n (µ). Since sup k I n k (µ k ) < ∞, we can apply the proposition 2.5.3 and in particular the equation (2.5.1): lim sup k→∞ α k π k (1/τ ) < ∞. Since ᾱ > 0, we have: lim sup k→∞ π k (1/τ ) < ∞. For k large enough and M large enough, we have

π k (1/τ ) ≤ M . As ν ∈ M 1 ((0, +∞] × {1, ..., n}), ν 1 τ ≤ r
is tight for all r > 0, there exists a subsequence of π k which converges in M 1 ((0, +∞]× {1, ..., n}). We can write the limit of π k (or its subsequence) as βπ+(1-β)δ (+∞) ⊗η 0 , for some β ∈ [0, 1] and η 0 ∈ M 1 ({1, ..., n}).

If β > 0, π(1/τ ) ≤ 1 β lim sup k→∞ π k (1/τ ) < ∞. If β = 0 we choose an arbitrary π such that π(1/τ ) < ∞.
Now, we prove that µ is in ∆ n , where µ is dened by:

µ = lim k µ k = lim k (α k µ 0,k + (1 -α k )δ (+∞,+∞) ⊗ η k ) = ᾱ lim k µ 0,k + (1 -ᾱ)δ (+∞,+∞) ⊗ (lim k η k ). It holds µ 0,k = [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) du ⊗ π k (dτ, dW ) -→ k→∞ β [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) du ⊗ π(dτ, dW ) + (1 -β)δ (+∞,+∞) ⊗ η 0 .
Let µ 0 and η be dened as:

µ 0 := [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) du ⊗ π(dτ, dW ) η := α(1 -β) 1 -ᾱβ η 0 + 1 - ᾱ 1 -ᾱβ (lim k η k ).
We get

µ = ᾱβµ 0 + ᾱ(1 -β)δ (+∞,+∞) ⊗ η 0 + (1 -ᾱ)δ (+∞,+∞) ⊗ (lim k η k ) = ᾱβµ 0 + (1 -ᾱβ)δ (+∞,+∞) ⊗ η,
and in particular µ ∈ ∆ n with α = β ᾱ.

Eventually

I n (µ) = β ᾱπ(1/τ )H(π|ψ n ) + (1 -β ᾱ)ξ = ᾱ [βπ(1/τ )H(π|ψ n ) + (1 -β)ξ] + (1 -ᾱ)ξ = ᾱπ k (1/τ )H(π k |ψ k ) + (1 -ᾱ)ξ + ᾱ [βπ(1/τ )H(π|ψ n ) + (1 -β)ξ -π k (1/τ )H(π k |ψ k )] ≤ lim inf k→∞ I n k (µ k ) + ᾱ [βπ(1/τ )H(π|ψ n ) + (1 -β)ξ -π k (1/τ )H(π k |ψ k )] .
In particular, we can apply Lemma 2.4 from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] (the proof can be directly adapted in our case -see 2.A.3), since the hypotheses therein are satised:

Lemma 2.5.8. Let π k ∈ M 1 ((0, +∞) × {1, ..., n}) be such that π k (1/τ ) < ∞ such that lim k π k (dτ, dW ) = βπ(dτ, dW ) + (1 -β)δ (+∞) (dτ ) ⊗ η 0 (dW ) (2.5.2) for some β ∈ [0, 1], π ∈ M 1 ((0, +∞) × {1, ..., n}) such that π(1/τ ) < ∞ and η 0 ∈ M 1 ({1, ..., n}). Then lim inf k→∞ π k (1/τ )H(π k |ψ k ) ≥ βπ(1/τ )H(π|ψ n ) + (1 -β)ξ. Eventually I n (µ) ≤ lim inf k→∞ I n k (µ k ) + ᾱ [βπ(1/τ )H(π|ψ n ) + (1 -β)ξ -π k (1/τ )H(π k |ψ k )] <0 by Lemma 2.5.8 ≤ lim inf k→∞ I n k (µ k ).
Remark 2.5.9. Once again we only used the fact that {1, ..., n} is compact so that we may replace it by any compact W. ♦

Proof of proposition 2.5.6. Let µ in M 1 ((0, +∞] 2 × {1, ..., n}) with I n (µ) < ∞. We want to prove that there exists a sequence

(µ k ) such that µ k µ, µ k ∈ ∆ n 0 for all k and lim sup k→∞ I n k (µ k ) ≤ I n (µ). Since I n (µ) < ∞, µ ∈ ∆ n . Let α, π and η corresponding to µ: µ(da, db, dc) =α [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) (da, db, dc)du ⊗ π(dτ, dW ) + (1 -α)δ (+∞,+∞) (da, db) ⊗ η(dc).
The aim of this (technical) proof is to construct a sequence of laws π k , depending on ψ k , which satises a condition on its limit (the condition (2.5.6) described below) and a condition on its entropy with respect to ψ k (the condition (2.5.7) described below). Then, we will construct a sequence of measure µ k which will satisfy the wished conditions. The next paragraphs give the details.

We denote ψ n (dτ ) the marginal law of τ in ψ n , and ψ k (dτ ) the corresponding law associated to ψ k to simplify notations. In particular, ψ n (dτ |W = j) is the marginal law of τ given W = j. We write q j the weight of j for η: q j = η({W = j}).

Fix ρ > 0, L > M > 1 such that P ψ n (τ = 1/M ) = P ψ n (τ = M ) = P ψ n (τ = L) = 0. Then there exist N ∈ N and 1/M = T 1 < T 2 < ... < T N = M such that T i+1 -T i ≤ ρ and P ψ n (τ = T i ) = 0.
Here of course N and T i depend on M and ρ. We also use the shorthand notation

A i = [T i , T i+1 ) and A = ∪ N i=1 A i in this proof. Then for L > M dene π ρ,L,M 0,k (dτ, dW ), η L,M k (dτ, dW ) and π ρ,M,L k (dτ, dW ) ∈ M 1 ((0, +∞) × {1, ..., n}) as: π ρ,M 0,k (dτ, dW ) = 1 β ρ,M k N i=1 n j=1 π(τ ∈ A i , W = j) π(A) τ ψ k (dτ, dW |τ ∈ A i , W = j), (2.5.3) η L,M k (dτ, dW ) = n j=1 1 ψ k (τ 1 τ ∈[M,L) |W = j) τ 1 τ ∈[M,L) q j ψ k (dτ |W = j) ⊗ δ j (dW ), (2.5.4) π ρ,L,M k (dτ, dW ) = απ ρ,M 0,k (dτ, dW ) + (1 -α)η L,M k (dτ, dW ), (2.5.5)
where β ρ,M k is the normalizing constant such that π ρ,M 0,k is a probability measure. Some q j 's can be equal to 0. In these cases, we consider in fact the sum on j such that the q j aren't zero: all the following calculation are then true. In (2.5.4), and in the following calculus, we consider the sum on i, j, such that the denominators are not null. To simplify the notation, we consider in the following calculus that η L,M k is the sum for j ∈ {1, ..., n}.

The above denition makes sense if L > M is large enough, and k is large enough depending on L and M (k will be sent to +∞ before L, and L before M ), and there is no problem with the conditioning. As ψ n (τ

∈ ∂A i ) = 0, if ψ k (A i × {j}) = 0 for each k large enough, then ψ(A i × {j}) = 0. Since I n (µ) < ∞, then H(π|ψ n ) < ∞ and π(A i × {j}) = 0. The term in (i, j) in (2.5.3) would be considered equal to 0. Similarly if ψ k ([M, L) × {n}) = 0 then the term in j in (2.5.4) vanishes.
If each terms of one sum is zero, then α is equal to 0 or 1, by the same arguments on the relative entropy.

We want to prove:

lim M →∞ lim L→∞ lim ρ→0 lim k→∞ π ρ,L,M k = απ + (1 -α)δ (+∞) ⊗ η, (2.5.6) lim inf M →∞ lim inf L→∞ lim inf ρ→0 lim sup k→∞ π ρ,L,M k (1/τ )H(π ρ,L,M k |ψ k ) ≤ I n (µ), (2.5.7) with I n (µ) = απ(1/τ )H(π|ψ n ) + (1 -α)ξ
, where the limits in M and L are understood to run over M and L satisfying the above conditions. Once (2.5.6) and (2.5.7) are proved, we can consider sequences

ρ k → 0, L k → ∞, M k → ∞ (such that ρ k , L k and M k satisfy the above conditions), we can dene π k = π ρ k ,L k ,M k k which satisfy: π k απ + (1 -α)δ (+∞) ⊗ η and lim sup k→∞ π k (1/τ )H(π k |ψ k ) ≤ I n (µ).
Then µ k dened by:

µ k (da, db, dc) := [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) (da, db, dc)du ⊗ π k (dτ, dW )
satises the proposition.

First, we prove the convergence (2.5.6) step by step. First, we consider πρ,M 0,k .

The normalisation constant β ρ,M k satises:

β ρ,M k = (0,+∞)×{1,...,n} N i=1 n j=1 π(τ ∈ A i , W = j) π(A) τ ψ k (dτ, dW |τ ∈ A i , W = j) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A) A i ×{j} τ ψ k (dτ, dW |τ ∈ A i , W = j) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A) ψ k (τ |τ ∈ A i , W = j) Since ψ k -→ k→∞ ψ, then ∀i ≤ N, ∀j ≤ n, ψ k (τ ∈ A i , W = j) -→ k→∞ ψ n (τ ∈ A i , W = j). By bounded convergence (because τ ∈ A i ⇒ τ ≤ T i+1 ), ∀j ≤ n, ∀f ∈ C b ((0, +∞) × {1, ..., n}), A i ×{j} τ f (τ, W )ψ k (dτ, dW ) -→ k→∞ A i ×{j} τ f (τ, W )ψ n (dτ, dW ).
So:

β ρ,M k → β ρ,M := N i=1 n j=1 π(τ ∈ A i , W = j) π(A) ψ n (τ |τ ∈ A i , W = j),
and

π ρ,M 0,k (dτ, dW ) →π ρ,M 0 (dτ, dW ), with π ρ,M 0 (dτ, dW ) := 1 β ρ,M N i=1 n j=1 π(τ ∈ A i , W = j) π(A) τ ψ n (dτ, dW |τ ∈ A i , W = j). Let f ∈ C b ((0, +∞) × {1, ..., n}). On A = [1/M, M ), for each j ≤ n, τ → τ f (τ, j) is uniformly continuous, so there exists a modulus of continuity ω f , such that ∀(x, y) ∈ A, ∀j ∈ {1, ..., n}, |xf (x, j) -yf (y, j)| ≤ ω f (|x -y|).
In fact, ω f is the maximum of the n modulus of continuity for functions τ → τ f (τ, j).

For all ρ > 0, (T i ) i dened as before, A×{1,...,n}

f (τ, W )τ π(τ ∈ A) π(dτ, dW ) = n j=1 A×{j} f (τ, j)τ π(τ ∈ A) π(dτ, dW ) = n j=1 N i=1 A i ×{j} f (τ, j)τ π(τ ∈ A) π(dτ, dW ).
Then:

n j=1 N i=1 A i ×{j} min τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(dτ, dW ) ≤ A×{1,...,n} τ f (τ, W ) π(τ ∈ A) π(dτ, dW ) ≤ n j=1 N i=1 A i ×{j} max τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(dτ, dW ) n j=1 N i=1 min τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(τ ∈ A i , W = j) ≤ A×{1,...,n} τ f (τ, W ) π(τ ∈ A) π(dτ, dW ) ≤ n j=1 N i=1 max τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(τ ∈ A i , W = j) Let us remark that n j=1 N i=1 max τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(τ ∈ A i , W = j) - n j=1 N i=1 min τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(τ ∈ A i , W = j) = 1 π(τ ∈ A) n j=1 N i=1 π(τ ∈ A i , W = j) max τ ∈A i (τ f (τ, j)) -min τ ∈A i (τ f (τ, j) ≤ 1 π(A) n j=1 N i=1 π(τ ∈ A i , W = j)ω f (ρ) = ω f (ρ) -→ ρ→0 0. Then n j=1 N i=1 min τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(τ ∈ A i , W = j) -→ ρ→0 A×{1,...,n} τ f (τ, W ) π(A) π(dτ, dW ) and n j=1 N i=1 max τ ∈A i (τ f (τ, j)) π(τ ∈ A) π(τ ∈ A i , W = j) -→ ρ→0 A×{1,...,n} τ f (τ, W ) π(A) π(dτ, dW ).
Then, by studying β ρ,M π ρ,M 0 (f ):

N i=1 n j=1 π(τ ∈ A i , W = j) π(A) min τ ∈A i (τ f (τ, j)) ≤ N i=1 n j=1 π(τ ∈ A i , W = j) π(A) τ f (τ, W )ψ n (dτ, dW |τ ∈ A i , W = j) ≤ N i=1 n j=1 π(τ ∈ A i , W = j) π(A) max τ ∈A i (τ f (τ, j)). So: ∀f ∈ C b ((0, +∞) × {1, ..., n}), β ρ,M π ρ,M 0 (f ) -→ ρ→0 A×{1,...,n} τ f (τ, W ) π(A) π(dτ, dW ).
(2.5.8)

In particular, for f = 1, π ρ,M 0

(1) = 1, and:

β ρ,M -→ ρ→0 β M := A×{1,...,n} τ π(A) π(dτ, dW ) = π(τ |τ ∈ A). Then π ρ,M 0 (dτ, dW ) π M 0 (dτ, dW ) := 1 β M τ π(dτ, dW |τ ∈ A). When M → ∞, π(τ ∈ A) = π(τ ∈ [ 1 M , M )) → 1. So: β M -→ M →∞ π(τ ) and: π M 0 (dτ, dW ) τ π(dτ, dW )/π(τ ) = π(dτ, dW ).
η L,M k (dτ, dW ) = n j=1 1 ψ k (τ 1 τ ∈[M,L) |W = j) τ 1 τ ∈[M,L) q j ψ k (dτ |W = j) ⊗ δ j (dW ).
We have ψ k (τ

1 τ ∈[M,L) |W = j) -→ k→∞ ψ n (τ 1 τ ∈[M,L) |W = j), so: η L,M k (dτ, dW ) → η L,M (dτ, dW ), with η L,M (dτ, dW ) := n j=1 1 ψ n (τ 1 τ ∈[M,L) |W = j) τ 1 τ ∈[M,L) q j ψ n (dτ |W = j) ⊗ δ j (dW ). When L → ∞, η L,M (dτ, dW ) → η M (dτ, dW ), with η M (dτ, dW ) := n j=1 1 ψ n (τ 1 τ ∈[M,+∞) |W = j) τ 1 τ ∈[M,+∞) q j ψ n (dτ |W = j) ⊗ δ j (dW ).
For g ∈ C b ((0, +∞) × {1, ..., n}) with a compact support,

η M (g) = n j=1 [M,∞) 1 ψ n (τ 1 τ ∈[M,+∞) |W = j)
τ g(τ, j)q j ψ n (dτ |W = j).

Thus, for M large enough, η M (g) = 0.

Moreover, for i ∈ {1, ..., n},

η M (1 W =i ) = [M,+∞) 1 ψ n (τ 1 τ ∈[M,+∞) |W = i) τ q i ψ n (dτ |W = i) = q i ψ n (τ 1 τ ∈[M,+∞) |W = i) ψ n (τ 1 τ ∈[M,+∞) |W = i) = q i . Then: η M (dτ, dW ) δ (+∞) (dτ ) ⊗ η(dW ). Now we deduce the convergence of π ρ,L,M k : lim M →∞ lim L→∞ lim ρ→0 lim k→∞ π ρ,L,M k = απ + (1 -α)δ (+∞) ⊗ η.
(2.5.6) is then proved.

πρ,L,M k (dτ, dW ) := 1 π ρ,L,M k (1/τ ) 1 τ π ρ,L,M k (dτ, dW ), (2.5.9) πρ,M 0,k (dτ, dW ) := 1 π ρ,M 0,k (1/τ ) 1 τ π ρ,M 0,k (dτ, dW ), (2.5.10) ηL,M k (dτ, dW ) := 1 η L,M k (1/τ ) 1 τ η L,M k (dτ, dW ).
(2.5.11)

In particular:

η L,M k (1/τ ) = n j=1 1 ψ k (τ 1 τ ∈[M,L) |W = j) τ τ 1 τ ∈[M,L) q j ψ k (dτ |W = j) ⊗ δ j (dW ) = n j=1 q j ψ k (1 τ ∈[M,L) |W = j) ψ k (τ 1 τ ∈[M,L) |W = j)   = n j=1 q j 1 ψ k (τ |τ ∈ [M, L), W = j)   ηL,M k (dτ, dW ) = 1 η L,M k (1/τ ) 1 τ n j=1 1 ψ k (τ 1 τ ∈[M,L) |W = j) τ 1 τ ∈[M,L) q j ψ k (dτ |W = j) ⊗ δ j (dW ) = 1 η L,M k (1/τ ) n j=1 1 ψ k (τ 1 τ ∈[M,L) |W = j) 1 τ ∈[M,L) q j ψ k (dτ |W = j) ⊗ δ j (dW ).
Using Lemma 2.3 from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] (the proof of which can be easily adapted in this situation

-see Lemma 2.A.2) , π → π(1/τ )H(π|ψ k ) = 1 π(τ ) H(π|ψ k ) is convex, so: π ρ,L,M k (1/τ )H(π ρ,L,M k |ψ k ) ≤ α 1 πρ,M 0,k (τ ) H(π ρ,M 0,k |ψ k ) + (1 -α) 1 ηL,M k (τ ) H(η L,M k |ψ k ).
First we consider the second term: since ψ n ({τ = M }) = ψ n ({τ = L}) = 0, we have:

lim k 1 ηL,M k (τ ) H(η L,M k |ψ k ) = 1 ηL,M (τ ) H(η L,M |ψ n ) ηL,M (τ ) = η L,M (1/τ ) -1 ηL,M (dτ, dW ) = 1 η L,M (1/τ ) n j=1 1 ψ n (τ 1 τ ∈[M,L) |W = j) 1 τ ∈[M,L) q j ψ n (dτ |W = j) ⊗ δ j (dW ).
We have

H(η L,M |ψ n ) = [M,L)×{1,...,n} ln ηL,M (dτ, dW ) ψ n (dτ, dW ) ηL,M (dτ, dW ) = 1 η L,M (1/τ ) n j=1 [M,L)×{j} ln ηL,M (dτ,dW ) ψ n (dτ,dW ) 1 τ ∈[M,L) q j ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (dτ |W = j) ⊗ δ j (dW ).

The functional I n 55

We can decompose ψ n in the following form:

ψ n (dτ, dW ) = n j=1 p j ψ n (dτ |W = j) ⊗ δ j (dτ ),
where p j = ψ n (W = j). By denition of ∆ n and {1, ..., n}, we know that p i = 0 implies q i = 0. Then:

H(η L,M |ψ n ) = 1 η L,M (1/τ ) × n j=1 [M,L)×{j} ln 1 η L,M (1/τ ) 1 τ ∈[M,L) q j ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (dτ |W = j) ⊗ δ j (dW ) p j ψ n (dτ |W = j) ⊗ δ j (dτ ) × 1 τ ∈[M,L) q j ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (dτ |W = j) ⊗ δ j (dW ) = 1 η L,M (1/τ ) n j=1 q j [M,L) ln q j p j η L,M (1/τ )ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (dτ |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) = 1 η L,M (1/τ ) n j=1 q j ln q j p j η L,M (1/τ )ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) = 1 η L,M (1/τ ) n j=1 q j ln q j p j ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) - 1 η L,M (1/τ ) n j=1 q j ln η L,M (1/τ )ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) .
Then we study

1 ηL,M k (τ ) H(η L,M k |ψ k ) = n j=1 q j ln q j p j ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) - n j=1 q j ln η L,M (1/τ )ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) .
First, we have ψ n (τ

1 τ ∈[M,L) |W = j) ≥ M , and ψ n (1 τ ∈[M,L) |W = j) ≤ 1, so ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) n j=1 q j ln q j p j ≤ 1 M n j=1 q j ln q j p j -→ M →∞ 0.

Chapter 2. LDP for cumulative processes

Secondly, we have:

n j=1 q j ln ψ n (η L,M (1/τ )τ 1 τ ∈[M,L) |W = j) ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) = n j=1 q j ln η L,M (1/τ ) ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) + n j=1 q j ln ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) = ln η L,M (1/τ ) × η L,M (1/τ ) + n j=1 q j ln ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (1 τ ∈[M,L) |W = j). Moreover, ψ n (τ 1 τ ∈[M,L) |W = j) ≥ M , so ψ n (τ 1 τ ∈[M,L) |W = j) -→ M,L→+∞ +∞. Since ln x x -→ x→+∞ 0 and ψ n (1 τ ∈[M,+∞) |W = j) -→ M →+∞
0, we have:

n j=1 q j ln ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (τ 1 τ ∈[M,L) |W = j) ψ n (1 τ ∈[M,L) |W = j) -→ M,L→∞ 0. Eventually, η L,M (1/τ ) = [M,L) 1 τ η L,M (dτ ) ≤ 1 M . Then in particular lim M →+∞ lim L→∞ η L,M 1 τ = 0.
(2.5.12)

Since x ln x -→ x→0 0, η L,M (1/τ ) × ln η L,M (1/τ ) -→ M,L→+∞ 0. Then lim M →∞ lim L→∞ H(η L,M |ψ n ) ηL,M (τ ) = 0 ≤ ξ.
We focus on the rst term. We know that πρ,M 0,k (τ ) = 1/π ρ,M 0,k (1/τ ). We have:

lim M →∞ lim L→∞ lim ρ→0 lim k→∞ π ρ,M 0,k (1/τ ) = lim M →∞ lim L→∞ lim ρ→0 1 β ρ,M N i=1 n j=1 π(τ ∈ A i , W = j) π(A) = lim M →∞ lim L→∞ lim ρ→0 1 β ρ,M = lim M →∞ lim L→∞ 1 β M = 1 π(τ )
.

(2.5.13)

Then:

1 πρ,M 0,k (τ ) → M,ρ,k 1 π(τ )
. Now, we study H(π ρ,M 0,k |ψ k ):

As π ρ,M 0 (1/τ ) = 1/β ρ,M , we have πρ,M 0 (dτ, dW ) = 1 π ρ,M 0 (1/τ ) 1 τ π ρ,M 0 (dτ, dW ) = β ρ,M × 1 β ρ,M N i=1 n j=1 π(τ ∈ A i , W = j) π(A) ψ n (dτ, dW |τ ∈ A i , W = j) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A) ψ n (dτ, dW |τ ∈ A i , W = j). lim k→+∞ H(π ρ,M 0,k |ψ k ) = H(π ρ,M 0 |ψ n ) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A) × ln N l=1 n m=1 π(τ ∈ A l , W = m) π(A) ψ n (dτ, dW |τ ∈ A l , W = m) ψ n (dτ, dW ) ψ n (dτ, dW |τ ∈ A i , W = j) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A) × A i ×{j} ln π(τ ∈ A i , W = j) π(A) ψ n (dτ, dW ) ψ n (τ ∈ A i , W = j)ψ n (dτ, dW ) ψ n (dτ, dW ) ψ n (τ ∈ A i , W = j) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A)ψ n (τ ∈ A i , W = j) × A i ×{j} ln π(τ ∈ A i , W = j) π(A)ψ n (τ ∈ A i , W = j) ψ n (dτ, dW ) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A) ln π(τ ∈ A i , W = j) π(A)ψ n (τ ∈ A i , W = j) = N i=1 n j=1 π(τ ∈ A i , W = j) π(A) ln π(τ ∈ A i , W = j) ψ n (τ ∈ A i , W = j) -ln(π(A)) = 1 π(A) N i=1 n j=1 π(τ ∈ A i , W = j) ln π(τ ∈ A i , W = j) ψ n (τ ∈ A i , W = j) -ln(π(A)) = 1 π(A) π(τ ∈ A c ) ln π(τ ∈ A c ) ψ n (τ ∈ A c ) + N i=1 n j=1 π(τ ∈ A i , W = j) ln π(τ ∈ A i , W = j) ψ n (τ ∈ A i , W = j)   -ln(π(A)) + π(A c ) ln π(A c ) ψ n (A c )
The well known decomposition of entropy property tells us that if (B m ) m≤M is a Chapter 2. LDP for cumulative processes partition of (0, +∞) × {1, ..., n} (explicitly proved in 2.B.1):

H(π|ψ n ) ≥ M m=1 π(B m ) ln π(B m ) ψ n (B m ) .
Here, (A i × {j}) i,j and A c are a partition of (0, +∞) × {1, ..., n}. We have:

lim k→+∞ H(π ρ,M 0,k |ψ k ) = H(π ρ,M 0 |ψ n ) ≤ 1 π(A) H(π|ψ n ) -ln(π(A)) + π(A c ) ln π(A c ) ψ n (A c ) . Since π(A) → 1 and π(A c ) → 0 as M → ∞: lim sup M →∞ sup ρ<1 lim k H(π ρ,M 0,k |ψ k ) ≤ H(π|ψ n ). Now, we can construct a sequence π k (= π ρ k ,L k ,M k k ) such that: π k απ + (1 - α)δ ∞ ⊗ η and : lim sup k→∞ π k (1/τ )H(π k |ψ k ) ≤ I n (µ).
Eventually, by (2.5.12) and (2.5.13),

µ k 1 a + b = π k 1 τ = απ ρ k ,M k 0,k 1 τ + (1 -α)η L k ,M k k 1 τ -→ k→∞ απ 1 τ = µ 1 a + b .
2.6 Some additional technical topological lemmata.

Before proving the upper bound and the lower bound of the LDP for empirical measures, we add two useful lemmata. The rst one will be used several times for the upper bound. Once again its proof uses conditioning.

Lemma 2.6.

1. Let D n M = µ ∈ ∆ n , µ 1 a + b ≤ M . Then D n M is a compact set.
Proof. Since D n M ⊂ X M we already know that it is relatively compact. But since ∆ n is not clearly closed, we have to show that D n M is closed. Let (µ k ) k be a sequence of measures in D n M . We will prove that some subse- quence again denoted (µ k ) k converges in D n M . We denote by α k , µ 0,k , π k and η k the corresponding quantities in the denition.

For any k, α k ∈ [0, 1] and η k ∈ M 1 ({1, ..., n}) which is compact, so there exists a subsequence α ϕ(k) , η ϕ(k) and α ∈ [0, 1] and η ∈ M 1 ({1, ..., n}) such that α ϕ(k) -→ k→∞ α and η ϕ(k) η.

We remind that, for any r > 0, ν ∈ M 1 ((0, +∞]), ν

1 x ≤ r is a compact set.
If α = 0 : then for f ∈ C b ((0, +∞] 2 × {1, ..., n}):

µ ϕ(k) (f ) = α ϕ(k) →0 µ 0,ϕ(k) (f ) ∈[-f ∞, f ∞] + (1 -α ϕ(k) ) →1 (δ (+∞,+∞) ⊗ η ϕ(k) η )(f ) -→ k→∞ (δ (+∞,+∞) ⊗ η)(f ). Since (δ (+∞,+∞) ⊗ η) ∈ D n M , µ ϕ(k) converges in D n M .
If α > 0 : then for k large enough, we have :

|α ϕ(k) -α| < α/2, so α ϕ(k) > α/2.
Then, we have:

π ϕ(k) 1 τ = µ 0,ϕ(k) 1 a + b = 1 α ϕ(k) µ ϕ(k) 1 a + b ≤ M α ϕ(k) ≤ 2M α .
We will prove that for all i, π ϕ(k)|W =i , which is the conditional law π given W = i, belongs to ν ∈ M 1 ((0, +∞]), ν 1 x ≤ r for a well-chosen r.

For any k, π ϕ(k) ∈ M 1 ((0, +∞] × {1, ..., n}), let dene (q j,ϕ(k) ) j∈{1,...,n} the weights such that q j,ϕ(k) = P π ϕ(k) (W = j). Then π ϕ(k) (1/τ ) can be written with its marginal law:

π ϕ(k) 1 τ = j∈{1,...,n} q j,ϕ(k) E π ϕ(k) 1 τ W = j .
The ((q j,ϕ(k) ) j∈{1,...,n} ) k are at most n sequences of weights in [0, 1]. There exists a subsequence (q j,ϕ 2 (k) ) j∈{1,...,n} and a (q j ) j∈{1,...,n} with q j ∈ [0, 1] such that q j = 1 and (q j,ϕ 2 (k) ) j∈{1,...,n} tends to (q j ) j∈{1,...,n} . Let J the set of indices j such that q j = 0. Let p min the minimum of the q j for j ∈ J:

p min = min{q j , j ∈ J} = 0.
Then, for k large enough, we have:

∀j ∈ J, q j,ϕ 2 (k) > p min 2 .
Then, we have, for each j ∈ J

π ϕ 2 (k) 1 τ = i∈{1,...,n} q i,ϕ 2 (k) E π ϕ 2 (k) 1 τ W = i ≥ p min 2 E π ϕ 2 (k) 1 τ W = j and π ϕ 2 (k) 1 τ ≤ 2M α , then: ∀j ∈ J, E π ϕ 2 (k) 1 τ W = j ≤ 4M αp min . So ∀j ∈ J, π ϕ 2 (k)|W =j ∈ ν ∈ M 1 ((0, +∞]), ν 1 x ≤ 4M/(αp min ) which is a com- pact set.
There exists a subsequence of (π ϕ 2 (k)|W =j ) k which converges to a π j for each j ∈ J. By a diagonal argument, we consider an extraction ϕ 3 such that:

∀j ∈ J, π ϕ 3 (k)|W =j π j .
We dene then π corresponding to (q j ) j and (π j ) j :

π(dτ, dW ) = j∈J q j δ (W =j) π j (dτ ) ∀f ∈ C b ((0, +∞] × {1, ..., n}), π(f ) = j∈J q j π j (f (., j)).
Then, π ϕ 3 (k) π.

It remains to show that µ to belong to D n M i.e to ∆ n . π can be written as:

π(dτ, dW ) = π(τ < +∞) × π(dτ, dW |τ < ∞) + (1 -π(τ < ∞))π(dτ, dW |τ = ∞). Let π * (dτ, dW ) = π(dτ, dW |τ < ∞), β * = π(τ < +∞) and η * such that π(dτ, dW |τ = ∞) = δ (+∞) ⊗ η * . Moreover, µ 0,ϕ(k)
is the product measure of du and π ϕ(k) . π ϕ 3 (k) weakly converges to π, so µ 0,ϕ 3 (k) weakly converges to the product measure of du and π. We denote by µ 0 the product measure of du and π * . Then µ ϕ(k) tends to

µ = αβ * µ 0 + α(1 -β * )δ (+∞,+∞) ⊗ η * + (1 -α)δ (+∞,+∞) ⊗ η. With η 0 = α(1 -β * ) (1 -αβ * ) η * + (1 -α) (1 -αβ * ) η, we can write µ = αβ * µ 0 + (1 -αβ * )δ (+∞,+∞) ⊗ η 0 , so µ ∈ ∆ n .
The next lemma introduces ∆n , a closed set which contains ∆ n (but is not its closure despite the notation). This will be used directly in the proof of the upper bound of the large deviation principle. A similar set is discussed in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] lemma 2.5.

However we have here to carefully manage the third coordinate, so that the proof in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] has to be rewritten.

Lemma 2.6.2. The set

∆n := { µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}) : µ = αµ 0 + (1 -α)δ (+∞,+∞) ⊗ η, α ∈ [0, 1], µ 0 (da, db, dc) = [0,1]×(0,∞)×{1,...,n} δ (uτ,(1-u)τ,W ) (da, db, dc)du ⊗ π(dτ, dW ), π ∈ M 1 ((0, +∞) × {1, ..., n}), η ∈ M 1 ({1, ..., n}) (2.6.1)
is closed in M 1 ((0, +∞] 2 × {1, ..., n}).

Proof. Let µ k ∈ ∆n such that µ k converges to a µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}) when k tends to ∞. We will prove that µ is in ∆n . Let α k , π k and η k be the corresponding quantities to µ k as in the denition of ∆n . (α k ) k ∈ [0, 1] admits a subsequence which converges to a α ∈ [0, 1]. η k is a sequence of measure on the nite set {1, ..., n}, so a subsequence of η k converges to η ∈ M 1 . As usual we identify the subsequence and the sequence. (π k ) k ∈ M 1 ((0, +∞) × {1, ..., n}) ⊂ M 1 ([0, +∞] × {1, ..., n}) also admits a subsequence which tends to a limit π ∈ M 1 ([0, +∞] × {1, ..., n}).

Then µ(da, db, dc) = α [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) (da, db, dc)du ⊗ π(dτ, dW ) + (1 -α)δ (+∞,+∞) (da, db) ⊗ η(dc).
We need to prove that µ is in ∆n . First we verify that π has no weight in 0, and secondly we prove that by rewritting µ, we can consider π ∈ M 1 ((0, +∞) × {1, ..., n}), ᾱ and η such that µ have the good form.

First, we consider the weight of 0 for π. By Skorohod's representation theorem, there exists a sequence (P k , V k ) k and X k of random variables such that (P k , V k ) ∈ (0, +∞) × {1, ..., n} has the law π k , X k has the law η k and (P k , V k ) converges a.s. to (P, V ) of law π, X k converges a.s. to X of law η. Let U be an uniform random variable independent of (P k ) k and P . For any f ∈ C b ([0, +∞] 2 ×{1, ..., n}) we obtain

µ k (f ) = α k E[f (U P k , (1 -U )P k , V k )] + (1 -α k )E[f (+∞, +∞, X k )] -→ k→∞ αE[f (U P, (1 -U )P, V )] + (1 -α)E[f (+∞, +∞, X)]
and this limit is equal to µ(f

). For f ε (a, b, c) = 1 a+b<ε , since µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}): απ({P < ε}) = µ(f ε ) -→ ε→0 0. Then π ∈ M 1 ((0, +∞] × {1, ...,

n}).

Now we prove that we can write µ in the same form that in ∆n . Let β = π(P = +∞).

If β = 0, then α [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,W ) (da, db, dc)du⊗π(dτ, dW )+(1-α)δ (+∞,+∞) (da, db)⊗η(dc) is already in the good form. Else, µ(f ) = αβE[f (U P, (1 -U )P, V )|P < ∞] + α(1 -β)E[f (+∞, +∞, V )|P = ∞] + (1 -α)E[f (+∞, +∞, X)] = αβE[f (U P, (1 -U )P, V )|P < ∞]+ (1 -αβ) α(1 -β) 1 -αβ E[f (+∞, +∞, V )|P = ∞] + 1 -α 1 -αβ E[f (+∞, +∞, X)] . Let η(dW ) = α(1-β) 1-αβ π(dW |τ = ∞)+
Lemma 2.6.4. If µ ∈ ∆ n then I n (µ) ≤ sup f ∈Γ µ(f ) and if µ ∈ ∆n \∆ n then sup f ∈Γ µ(f ) = +∞. (recall that Γ is dened in (2.4.9))
The proof of the Lemma is almost identical to [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] (see Lemma 2.A.5 and Lemma 2.A.6). Indeed it is enough to consider a third variable c ∈ {1, ..., n}. Similarly to the proof of Lemma 2.6 we have to introduce a function

f d,ϕ,M (a, b, c) = ϕ(a + b, c) a + b + d1 (M,+∞] (a + b)
for a continuous and compactly supported ϕ. f d,ϕ,M is lower semi-continuous. One can then copy the proof of Lemma 2.6 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. For the second part we may consider the same function f ε as in the proof of Lemma 2.7 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF].

Notice that again we may replace {1, ..., n} by a compact W.

The second one (and its proof ) is identical to Proposition 3.3 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] (see Proposition 2.A.7) Lemma 2.6.5. For all f ∈ Γ and all t > 0, E(e t µ n

t (f ) ) ≤ D n,f 1-C n,f < +∞.
2.7 Proof of Theorem 2.4.3.

In this section, we prove the LDP for P n t that denotes the P distribution of µ n t .

This time the introduction of the third coordinate W replacing F (τ ) does not create any new diculty. However some points in the proofs of [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] are not clear for us and we will give the details for some points.

The proof of the upper bound is made in several steps: the proof of a weak principle, for compact sets C (itself divided in several steps), and the proof that µ n t is an exponentially tight family, i.e. satises: for all α ∈ R + there exists some compact set K α with lim sup

t→+∞ 1 t ln P n t (K c α ) < -α .
The (full) upper bound for closed sets C then follows from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. These steps are described in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] section 3, and are given in Appendix 2.A.3.1.

Exponential tightness is the aim of Lemma 3.1 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. The rst step is the following lemma Lemma 2.7.1. We have: The proof of the upper bound for compact subsets is done in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] p. 2261 and 2262 beginning with the denition of the set

lim M →+∞ lim sup t→+∞ 1 t ln P µ n t 1 a + b > M = -∞.
∆ M,g,δ = {µ ∈ M 1 ((0, +∞] 2 ) , ∃ν ∈ ∆, |ν(g) -µ(g)| ≤ δ, µ(1/(a + b)) ≤ M }
for some continuous and bounded g, δ and M positive. We do not see an immediate argument showing that this set is closed (hence compact since it is relatively compact). We will thus slightly modify the proof in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF].

We introduce a modied set for M δ and g as before,

∆ n M,g,δ = {µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}) , ∃ν ∈ ∆ n , |ν(g) -µ(g)| ≤ δ, µ(1/(a + b)) ≤ M, ν(1/(a + b)) ≤ M + δ} .
(2.7.1)

We also dene

R n M,g,δ := -lim sup t→+∞ 1 t ln P n t ((∆ n M,g,δ ) c ) . (2.7.2)
One can of course replace g by a nite number of continuous and bounded g i 's.

Lemma 2.7.2. ∆ n M,g,δ is a compact set.

Proof. Notice that if µ k ∈ ∆ n M,g,δ weakly converges to some µ, µ ∈ D n M according to lemma 2.6.1. The corresponding sequence ν k in ∆ n actually belongs to D n M +δ so that one can nd a subsequence still denoted ν k that converges to some ν ∈ D n M +δ (this is the key dierence with ∆ M,g,δ ). Since g is bounded and continuous, by taking limits we have |ν(g) -µ(g)| ≤ δ. Of course for ε > 0,

ν(1/(a + b + ε)) = lim k ν k (1/(a + b + ε)) ≤ lim k ν k (1/(a + b)) ≤ M + δ
ν n t (f ) := 1 t Mt i=1 τ i 1 0 f (uτ i , (1 -u)τ i , W n i )du + t -S Mt t f (+∞, +∞, W n Mt+1 ) .
Then ν n t ∈ ∆ n almost surely. For all g, δ there exists some t(g, δ) such that for t ≥ t(g, δ), the events {|µ n t (g)ν n t (g)| > δ} and { µ n Proof. This lemma is the analogue of Lemma 3.2 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] but due to the modication of our ∆ n M,g,δ we have to complete the proof therein.

Arguing as in the proof of Lemma 3.2 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] one shows that for t large enough the set {|µ n t (g) -ν n t (g)| > δ} is almost surely empty (see Lemma 2.A.9). For the other term,

µ n t 1 a + b -ν n t 1 a + b = 1 t Mt i=1 τ i × 1 τ i + τ Mt+1 t × t -S Mt τ Mt+1 × 1 τ Mt+1 - 1 t Mt i=1 τ i × 1 τ i = 1 t t -S Mt τ Mt+1 ≤ 1 t .
Hence for t > 1/δ we have µ n t

1 a+b -ν n t 1 a+b ≤ δ. Corollary 2.7.4. lim M →+∞ R n M,δ,g = +∞.
Proof. According to lemma 2.7.3, for t large enough (that does not depend on M but only on δ and g) the set {µ n t (1/(a+b)) ≤ M }∩(∆ n M,g,δ ) c is almost surely empty. Hence for t large enough

P n ((∆ n M,g,δ ) c ) ≤ P n (µ n t (1/(a + b)) > M )
and the result follows from lemma 2.7.1.

We may now follow the proof in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. First exactly as in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF], thanks to Lemma 2.6.5, for all open set O, all g, M , δ and all f ∈ Γ, lim sup

t→+∞ 1 t ln P n t (O) ≤ -inf ν∈O I n f,M,g,δ (µ)
where

I n f,M,g,δ (µ) = µ(f ) ∧ R n M,g,δ if µ ∈ ∆ n M,g,δ +∞ otherwise. , (2.7.3)
Since f is lower semicontinuous, I n f,M,g,δ (µ) is also lower semicontinuous thanks to the compactness of ∆ n M,g,δ . One can thus deduce as in [52] that for all compact subset K, lim sup

t→+∞ 1 t ln P n t (K) ≤ -inf µ∈K sup f,M,g,δ I n f,M,g,δ (µ) := -inf µ∈K Ĩ(µ) .
If µ / ∈ ∆n which is closed according to lemma 2.6.2, one can nd an open neighbor- hood U of µ such that U ∩ ∆n = ∅. We may choose U of the form

U = k i=1 {ν, |ν(g i ) -µ(g i )| ≤ δ}
for some family g 1 , ..., g k of bounded and continuous functions, so that for at least one of the g i 's denoted by g, µ / ∈ ∆ n M,g,δ . Hence ∩ g,δ ∆ n M,g,δ ⊂ ∆n and Ĩ(µ) = +∞ if µ / ∈ ∆n . Together with corollary 2.7.4 we deduce that

Ĩ(µ) ≥ sup f ∈Γ I f (µ) where I f (µ) = µ(f ) if µ ∈ ∆n , I f (µ) = +∞ otherwise.
Thus, according to lemma 2.6.4, Ĩ(µ) ≥ I n (µ) and the upper bound is proved.

The proof of the lower bound is similar to [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF], just replacing the sample τ i by a sample (τ i , W i ), and thus is omitted. (See Appendix 2.A.3.2 for the totality of the proof.) 2.8 Proof of Theorem 2.4.4.

We turn to the proof of theorem 2.4.4. We rst will deduce a LDP from theorem 2.4.3 by using the contraction principle, and then identify the rate function.

The contraction principle.

Dene ϕ(a, b, c) = c a+b which is continuous on (0, +∞] 2 × {1, ..., n}. Then

µ n t (ϕ) = 1 t Mt i=1 W n i + (t -S Mt )W n Mt+1 t τ Mt+1 .
Remark that

(t -S Mt )W n Mt+1 t τ Mt+1 ≤ n t so that lim sup t→∞ 1 t ln P µ n t (ϕ) - 1 t Mt i=1 W n i > δ = -∞
showing that µ n t (ϕ) and 1 t Mt i=1 W n i = Z n t /t will satisfy the same LDP. The contraction principle (Theorem 4.2.1 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]) should thus furnish some LDP for Z n t /t. Unfortunately ϕ is not bounded so that µ → µ(ϕ) is not continuous from M 1 (0, +∞] 2 × {1, ..., n}) to R and the contraction principle does not apply directly. We are thus obliged one more time to use an approximation procedure replacing ϕ by

ϕ ε (a, b, c) = c (a + b) ∨ ε for ε > 0. Replacing τ i by τ ε i = τ i ∨ ε we may introduce M ε t , A ε t , B ε t , C n,
J ε,n (m) = inf{I n,ε (µ) , µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), m = µ(ϕ ε )} .
(2.8.1)

Chapter 2. LDP for cumulative processes

We now have

µ ε,n t (ϕ ε ) = 1 t M ε t i=1 W n i + (t -S ε M ε t )W n M ε t +1 t τ ε M ε t +1 , so that, since M ε t ≤ M t , µ ε,n t (ϕ ε ) - 1 t Mt i=1 W n i ≤ 1 t Mt i=M ε t +1 W n i + (t -S ε M ε t )W n M ε t +1 t τ ε M ε t +1 ≤ n t ((M t -M ε t ) + 1
Jn (m) = sup δ>0 lim inf ε→0 inf |z-m|<δ J ε,n (m) . (2.8.2)
In particular we know that Jn is l.s.c. so that its level sets are closed.

Study of the rate function.

The goal of this subsection is to show the following lemma (partly close to Lemma 5.1 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]) explaining the various forms of the rate function (recall that ϕ(a, b, c) = c/(a + b)).

Lemma 2.8.1. We dene, for all m > 0, Jn (m

) := inf{I n (µ), µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), m = µ(ϕ)}. Then i) Jn = J n (J n is dened in (2.2.6)). In addition Jn (m) = inf{I n 0 (µ), µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), m = µ(ϕ)}.
ii) We also have:

Jn (m) = min{I n (µ), µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), m = µ(ϕ)} (2.8.3) = min{I n 0 (µ), µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), m = µ(ϕ)}. (2.8.4)
iii) Finally,

J n = Jn = Jn . Proof. Proof of i) We have Jn (m) := inf I n (µ), µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), c a + b µ(da, db, dc) = m = inf I n (µ), µ ∈ ∆ n , c a + b µ(da, db, dc) = m = inf απ 1 τ H(π|ψ n ) + (1 -α)ξ, α ∈ [0, 1], µ 0 ∈ ∆ n 0 , α c a + b µ 0 (da, db, dc) = m = inf απ 1 τ H(π|ψ n ) + (1 -α)ξ, α ∈ [0, 1], µ 0 ∈ ∆ n 0 , απ W τ = m = inf α 1 π(τ ) H(π|ψ n ) + (1 -α)ξ, α ∈ [0, 1], π ∈ M 1 ((0, +∞) × {1, ..., n}), α π(W ) π(τ ) = m = inf α β H(π|ψ n ) + (1 -α)ξ, α ∈ [0, 1], β > 0, π(τ ) = β, α π(W ) β = m Let p(a, b) = inf{H(ν|ψ n ), ν(τ ) = a, ν(W ) = b}. We have p = Λ * n according to
Csiszar I-projection theorem (Theorem 3 in [START_REF] Csiszar | Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem[END_REF]). As in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] (proof of Lemma 5.1)

another way is to directly prove the dual equality p * = Λ n .

We thus have

Jn (m) = inf α β Λ * n β, mβ α + (1 -α)ξ, α ∈ [0, 1], β > 0 . But α β Λ * n β, mβ α = sup x,y αx + my - α β Λ n (x, y) = β Λ * n α β , m β where β = α β . Thus Jn (m) = inf βΛ * n α β , m β + (1 -α)ξ, α ∈ [0, 1], β > 0 .
We will show that :

inf α∈[0,1] βΛ * n α β , m β + (1 -α)ξ = βΛ * n 1 β , m β .
Taking α = 1, we see that the left hand side is less than or equal to the right hand side. To show the converse inequality, pick α ∈ [0, 1]:

βΛ * n α β , m β + (1 -α)ξ = sup x,y {αx + (1 -α)ξ + my -βΛ n (x, y)} ≥ sup x,y {x ∧ ξ + my -βΛ n (x, y)}.
Since W is bounded, e yW ≥ C(y) > 0 for all y, so that we have for all x > ξ and all y, ψ n (e xτ +yW ) ≥ C(y) ψ n (e xτ ) = +∞ .

This shows that Λ n (x, y) = +∞, for all x > ξ and for all y. Hence, the supremum on x can be restricted to the supremum on {x ≤ ξ}:

βΛ * n α β , m β + (1 -α)ξ ≥ sup x,y {x ∧ ξ + my -βΛ n (x, y)} = sup x≤ξ,y {x + my -βΛ n (x, y)} = βΛ * n 1 β , m β
and the desired inequality is proved.

Notice that during the proof we have seen than the minimization is obtained looking only at µ ∈ ∆ n 0 so that we may replace I n by I n 0 in the denition of Jn .

Proof of ii). This part is completely similar to the corresponding one in the proof of Lemma 5.1 in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. The only thing to see is that we may replace ||F || ∞ in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] by

K if |W | ≤ K.

Proof of iii).

To prove this equality, we rst prove the inequality Jn (m) ≤ Jn (m).

Let m ∈ R + and ε > 0. First, remark that Jn (m) ≤ J n,ε (m). Indeed, if µ ∈ ∆ n , with associated π, α and η, is such that H(π|ψ n,ε ) < +∞ and µ(ϕ ε ) = m, then π has its support included in (ε, +∞) × {1, ..., n}. Hence I n (µ) = I n,ε (µ). Moreover, µ(ϕ) = µ 1 a+b = m. This yields

Jn (m) = inf{I n (µ), µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), m = µ(ϕ)} ≤ inf{I n (µ), µ ∈ M 1 ({(a, b, c) ∈ (0, +∞) 2 × {1, ..., n}), µ({a + b ≤ ε}) = 0, m = µ(ϕ)} = inf{I n,ε (µ), µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}), m = µ(ϕ ε )} = J n,ε .
We will take the limit as ε → 0. To this end we may write for δ > 0, inf z,|z-m|≤δ

Jn (z) ≤ inf z,|z-m|≤δ J n,ε (z) so that inf z,|z-m|≤δ Jn (z) ≤ lim inf ε→0 inf z,|z-m|≤δ J n,ε (z) ≤ Jn (m)
Since Jn = J n , it is clearly lower semicontinuous so that

Jn (m) ≤ lim inf δ→0 inf z,|z-m|≤δ Jn (z) ≤ Jn (m) .
Finally Jn ≤ Jn Now, we prove that Jn ≥ Jn . Since Jn (m) is a minimum, let μ a measure such that Jn (m) = I n (μ) and μ(ϕ) = m. For a sequence ε k going to 0, we consider ψ k the distribution of (τ ∨ ε k , W ) . Using proposition 2.5.6, there exists µ k ∈ ∆ n 0 with associated to ψ k such that µ k μ and lim sup k→∞ I n k (µ k ) ≤ I n (μ). Here I n k = I n , so lim sup k→∞ I n (µ k ) ≤ I n (μ).

By construction µ k ({a + b < ε k }) = 0 and for πk associated to µ k , we have

H(π k |ψ n ) < +∞. Since πk is a measure on (ε k , +∞) × {1, ..., n}, H(π k |ψ k ) = H( πk |ψ n ), and it follows I n (µ k ) = I n,ε k (µ k ).
As a consequence of proposition 2.5.6 it holds µ k

1 a+b → μ 1 a+b . Hence, δ k = µ k 1 a+b -μ 1 a+b → 0. We can now write inf z,|z-m|≤δ k J n,ε k (z) ≤ I n,ε k (µ k ) = I n (µ k ) so that lim sup k→∞ inf z,|z-m|≤δ k J n,ε k (z) ≤ lim sup k→∞ I n (µ k ) ≤ I n (μ) = Jn (m)
and nally At this point we have obtained that Z n t /t satises a weak LDP with rate function J n . In order to get the full LDP we have to show that the level sets are bounded (since we know that they are closed). Recall that it is not a direct consequence of the contraction principle since µ → µ(ϕ) is not continuous.

Jn (m) = sup δ>0 lim inf ε→0 inf z,|z-m|<δ J n,ε (z) ≤ Jn (m) .
First since J n ≤ J n,ε for all ε > 0, for all closed set F it holds lim sup

ε→0 inf y∈F J n,ε (y) ≥ inf y∈F J n (y).
It remains to show that J n is a good rate function i.e. that it has compact level sets. According to the previous subsection for a sequence m k such that J n (m k ) ≤ β, 2.9 Proof of Theorem 2.2.5.

When W is bounded and discrete, the full LDP is already given by Theorem 2.4.4. If W isn't bounded, or isn't discrete, we need to dierentiate whether β 0 = ∞ or not.

In the rst case, when β 0 = ∞, we are able to prove a Large Deviation principle. Since the rate function J is dicult to calculate, we compare J with an other function J to simplify some inequalities. This work is done in the following subsection 2.9.1.

In the second case, since the approximation Z n t /t isn't an exponentially good approximation of Z t /t, we only prove some useful inequalities of deviations, but we're not able to prove the Large Deviation Principle.

Case A:

β 0 = ∞ In this case, Z n t /
J(m) = sup δ>0 lim inf n→∞ inf |m-z|<δ J n (z) .
In order to obtain a full LDP, we use lemma 2.2.3. Therefore it remains to show that Z t /t is exponentially tight. Lemma 2.9.1. Assume that Assumption 2.2.4 is fullled, then (Z t /t) t≥0 is exponentially tight, i.e. for all α > 0, there exists a compact set K α such that lim sup

t→+∞ 1 t ln P Z t t / ∈ K c α < -α.
Proof. Since Z n t /t is an approximation of Z t /t and satises a full LDP, we can decompose the probability as following: for each n, and for all δ: (2.9.2) (If β 0 = +∞, we can consider all δ > 0.)

P Z t t / ∈ [-A, A] ≤ P Z t t - Z n t t > δ + P Z n t t / ∈ [-A + δ, A -δ] ≤ P Z t t - Z n t t > δ + P Z n t t < -A + δ + P Z n t t > A -δ . ≤ 3 max P Z t t - Z n t t > δ , P Z n t t < -A + δ , P Z n t t > A -δ . ( 2
We just have to study P Z n t t > A -δ and the symmetric case. We know from Theorem 2.4.4 that:

lim t 1 t ln P Z n t t > B ≤ -inf m>B J n (m).
It remains to show that ∀α > 0 one can choose a level B α such that ∀m > B α , J n (m) > α.

Remind that

J n (m) = inf β>0 βλ * n 1 β , m β = inf β>0 sup x,y x + my -β ln E[e xτ +yWn ] ,
(where x + my -

W n = W ∧ n ∨ (-n)). Since -|W | ≤ W n ≤ |W |,
β 2 ln E[e xτ +y|W | ] 2 ≥ sup x∈R,y≥0 x + my - β 2 ln E[e 2xτ ] - β 2 ln E[e 2y|W | ] ≥ sup x∈R x - β 2 ln E[e 2xτ ] + sup y≥0 my - β 2 ln E[e 2y|W | ]
There exists x 0 < 0 such that E[e 2x 0 τ ] ≤ e -1 , and since β 0 > 0, y → E[e y|W | ] is strictly increasing and continuous on [0, β 0 ). Then there exists y 0 > 0 such that 72 Chapter 2. LDP for cumulative processes

1 < E[e 2y 0 |W | ] ≤ e. Therefore sup x∈R,y≥0 x + my - β 2 ln E[e xτ +y|W | ] 2 ≥ x 0 - β 2 ln E[e 2x 0 τ ] + my 0 - β 2 ln E[e 2y 0 |W | ] ≥ x 0 + my 0 + β 2 1 -ln E[e 2y 0 |W | ] Finally, since 1 -ln E[e 2y 0 |W | ] > 0, we conclude that J |.| (m) ≥ x 0 + my 0 -→ m→∞ +∞.
In conclusion, let α > 0 and δ > 2α β 0

(or δ > 0 if β 0 = +∞), let n(α, δ) dened in (2.9.2) and n ≥ n(α, δ), let A = α-x 0 y 0 + δ (x 0 and y 0 are determined by the law of τ and W ). We have

lim t 1 t ln P Z n t t > A -δ ≤ -inf m>A-δ J n (m) ≤ -inf m>-A+δ J |.| (m) ≤ -inf m>A-δ x 0 + my 0 ≤ -x 0 -y 0 (A -δ) ≤ -α
and using the evenness of

J |.| , lim t 1 t ln P Z n t t < -A + δ ≤ -α.
Eventually combining these bounds with (2.9.2) in (2.9.1), and deduce that for n ≥ n(α, δ),

lim t 1 t ln P Z t t / ∈ [-A, A] ≤ -α.
Now, we have a full LDP for Z t /t, but the expression of J isn't very convenient.

In particular, this expression can depend on the reduction used. In order to simplify some inequality and obtain Equations (2.2.8) and (2.2.9), we prove J ≥ J dened in (2.2.6).

We consider two cases: in the rst one, Lemma 2.9.2, we assume W is bounded by K and consider the discretization W n dened in subsection 2.3.2. Then in the second one, Lemma 2.9.5, we will examine the general case.

Contrary to [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] we do not know a priori that J is a good rate function. However, we use results of Lemma 2.8.1 replacing {1, ..., n} by [-K, K]. In particular we deduce that

J(m) = J(m) := min{I(µ) ; µ ∈ M 1 ((0, +∞] 2 × [-K, K]) , µ(ϕ) = m} .
We want to prove Lemma 2.9.2. If W is bounded, it holds J ≤ J. Proof. We may of course assume that J(m) < +∞. One can thus nd sequences

m n and ε n ≤ 1 such that ε n → 0, m n → m and J n (m n ) ≤ J(m) + ε n . Since J n = Jn , one can thus nd a sequence µ n ∈ M 1 ((0, +∞] 2 × [-K, K]) such that I n (µ n ) < J(m) + 1.
According to proposition 2.5.3 (recall that it is true here, see remark 2.5.5) one can thus nd some subsequence still denoted by µ n converging to some µ in M 1 ((0, +∞] 2 × [-K, K]). Now (again recall remark 2.5.5) according to proposition 2.5.4 we have lim inf n I n (µ n ) ≥ I(µ) so that I(µ) ≤ J(m).

In order to see that I(µ) ≥ J(m) it remains to show that µ(c/(a + b)) = m and to apply J = J.

Notice that thanks to ii) in lemma 2.8.1 we may have chosen µ n ∈ ∆ n 0 so that We will deduce that πn is tight. Indeed a standard application of the Orlicz-Hölder inequality (with the conjugate pair u → e u -1 -u and u → u → u ln u -u)

shows that πn (A) ≤ κ C 1 ln(1/ψ n (A))
for some universal constant κ. Choosing A = {a + b < ε} so that ψ n (A) = ψ(A) we get the desired result choosing ε small enough.

It follows

µ n c a + b ε -1 (a + b)1 a+b≤ε + 1 a+b>ε -m n ≤ K µ n (1/(a + b)) Cκ µ n ({a + b ≤ ε}) ≤ C µ({a + b ≤ ε}) .
Since the integrated function is bounded and continuous we can pass to the limit in n rst, and then in ε using Lebesgue's bounded convergence theorem since

c a + b ε -1 (a + b)1 a+b≤ε + 1 a+b>ε ≤ K a + b
which is µ integrable according to what we did before. This nally shows that µ(c/(a + b)) = m and concludes the proof.

Chapter 2. LDP for cumulative processes

We do not know about the converse inequality. Remark 2.9.3. We will see below another way to prove this result (in an even more general context). Nevertheless we have given this proof in order to complete the picture in the [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] context.

♦

We will now directly study the function J. Recall that We do not assume here that W is bounded. Lemma 2.9.4. J is a good rate function.

J(m) = inf
Proof. First we remark that

sup x,y Λ(m, β, x, y) ≥ sup x Λ(m, β, x, 0) = sup x (x -β ln E(e xτ )) .
Since τ ≥ 0 (and supposed not to be identically 0) one can nd x τ < 0 such that E(e xτ τ ) = e -1 (we already use this in the proof of lemma 2.3.1), so that sup x,y Λ(m, β, x, y) ≥ x τ + β .

Let {J ≤ M } be some level set of J. where 0 ≤ κ < 1 ∧ β 0 , β 0 being dened in Assumption 2.2.4 ii). For β ≤ β τ , β ln E(e κW ) is thus bounded by C, so that

J(m) ≥ mκ -C showing that {J ≤ M } is bounded.
It remains to show that the level sets are closed. Let m n → m be a sequence in {J ≤ M }. According to what precedes we know that the inmum in β has to be taken in a bounded interval, so that one can nd a (sub)-sequence β n converging to β such that J(m n ) ≤ sup

x,y Λ(m n , β n , x, y) + ε n with ε n going to 0. This implies that for all (x, y)

x + m n y -β n ln E(e xτ +yW ) ≤ M + ε n which implies J(m) ≤ M by taking the limit in n and then the supremum w.r.t. (x, y).

2.9. Proof of Theorem 2.2.5.

75

For a general W , consider

J(m) = sup δ>0 lim inf n→∞ inf |m-z|<δ
J n (z) .

We now know that Z n t /t satises a full LDP with good rate function J n . We may state Lemma 2.9.5. It holds J ≤ J.

Proof. As for the proof of lemma 2.9.2, we may of assume that J(m) < +∞ and thus nd sequences m n and ε n ≤

1 such that ε n → 0, m n → m and J n (m n ) ≤ J(m) + ε n . We denote δ n = |m -m n |. Dene M = J(m) + 1.
For n large enough, m n belongs to {J n ≤ M }. We have seen in the proof of lemma 2.9.4 that J n (m n ) is thus given by the inmum for 0 < β ≤ M -|x τ | = β τ where E(e xτ τ ) = e -1 , i.e x τ does not depend on n. The same holds with J(m). If β ∈ (0, β τ ] and ε > 0 we may found

(x ε , y ε ) such that βΛ * (1/β, m/β) ≤ x ε + my ε -β ln E e xετ +yεW + ε ≤ x ε + m n y ε -β ln E e xετ +yεW + ε + δ n |y ε | . Since E e xετ +yεW n → E e xετ +yεW
as n growths to innity, for n large enough the dierence is less than ε so that

βΛ * (1/β, m/β) ≤ x ε + m n y ε -β ln E e xετ +yεW n + (1 + β τ )ε + δ n |y ε | ≤ βΛ * n (1/β, m n /β) + (1 + β τ )ε + δ n |y ε | .
Taking the inmum in β and then the lim inf n we get

J(m) ≤ lim inf n J n (m n ) + (1 + β τ )ε ≤ J(m) + (1 + β τ )ε.
It remains to let ε go to 0 to conclude.

Remark 2.9.6. When W = 1 a.s. it is easily seen that sup x,y Λ(m, β, x, y) = +∞ except for m = β yielding J(m) = sup x (x -m ln E(e xτ )) as expected.

Since J is dened on R one can expect some monotonicity on intervals delimited by the asymptotic mean E(W )/E(τ ). We were not able to prove this monotonicity. ♦ 2.9.2 Case B: β 0 < ∞ We remind that W n is a reduction of W such that Z n t /t is an approximation of Z t /t and satises a full LDP, as proved in Theorem 2.4.4. However, this approximation is not an exponentially good approximation, therefore the LDP cannot be transferred to Z t /t. In this case, we prove the deviation Inequalities (2.2.10) and (2.2.11). We will focus on the rst one, P Zt t ≥ m + a , since the proof is exactly Chapter 2. LDP for cumulative processes the same for the second one, P Zt t ≤ m -a .

For each n ∈ N * , for κ ∈ (0, 1), we have

P Z t t ≥ m + a ≤ P Z n t t ≥ m + κa + P Z t t - Z n t t ≥ (1 -κ)a ≤ 2 max P Z n t t ≥ m + κa , P Z t t - Z n t t ≥ (1 -κ)a
Then, for each n ∈ N * and κ ∈ (0, 1),

t ≥ m + a ≤ max lim inf n→∞ -inf z≥m+κa J n (z) , -β 0 (1 -κ)a 2 ≤ -min lim sup n→∞ inf z≥m+κa J n (z), β 0 (1 -κ)a 2 .
Since this inequality is satised for each n, we can then apply the Lemma 2.9.8, proved below, which gives us lim sup

n→∞ inf z≥m+κa J n (z) ≥ inf z≥m+κa J(z),
and we obtain the Inequation (2.2.10): for all κ ∈ (0, 1) lim sup

t→∞ 1 t ln P Z t t ≥ m + a ≤ -min inf z≥m+κa J(z), β 0 (1 -κ)a 2 .
Remark 2.9.7. Notice J n isn't the same in each case. J n can be written

J n (z) = inf β>0 sup x,y {x + my -β ln E e xτ +yW n },
but the denition of W n depends on the reduction: If W is only bounded by K, a reduction with nite valued W n was necessary:

W n = n-1 j=-n jK n 1 W ∈[jK/n,(j+1)K/n[ .
If W is only discrete, a reduction with W n bounded by -n and n have been done:

W n = (W ∨ n) ∧ (-n).
If W isn't bounded nor discrete, another reduction is necessary:

W n = -n1 W <-n + n1 W ≥n + n 2 -1 j=-n 2 j n 1 W ∈[ j n , j+1 n ) .
Since in each case, W n is discrete and bounded a.s. and E e xτ +yW n → n→∞ E e xτ +yW for (x, y) such that E e xτ +yW < ∞, the proof is the same.

Lemma 2.9.8. Assume (W n ) n is a sequence of random variables which converges almost surely to W and such that for each n, W n is discrete and bounded a.s. Then, we have, for all z 0 ∈ R,

lim sup n→∞ inf z≥z 0 J n (z) ≥ inf z≥z 0 J(z),
where J n is dened by

J n (z) = inf β>0 sup x,y {x + my -β ln E e xτ +yW n }.
Proof. Let (W n ) n be a sequence of random variables which converges almost surely to W , such that for each n, W n is discrete and bounded a.s. Let z 0 ∈ R. Thanks to Lemma 2.8.1, J n has dierent forms. Here we will use the form

J n (z) = inf β>0 βΛ * n 1 β , m β .
If lim sup n→∞ inf z≥z 0 J n (z) = ∞, then the inequality is satised. Let assume lim sup n→∞ inf z≥z 0 J n (z) < ∞. We denote C(z 0 ) this quantity. There exists a sequence n k → ∞ such that lim k→∞ inf z≥z 0 J n k (z) = C(z 0 ). For each k ∈ N, there exists a sequence (

z k i ) i in [z 0 , +∞) such that inf z≥z 0 J n k (z) = lim i J n k (z k i ). Let δ > 0. There exists K such that ∀k ≥ K, inf z≥z 0 J n k (z) ∈ [C(z 0 ) -δ/2, C(z 0 ) + δ/2] and I k such that ∀i ≥ I K , J n k (z k i ) ∈ [C(z 0 ) -δ, C(z 0 ) + δ]. Then, for k ≥ K, and i ≥ I k , z k i is in {J n k ≤ C(z 0 ) + δ}. Moreover, for each n ∈ N , J n ≥ J |.
| by the proof of the Lemma 2.9.1, where

J |.| (m) := inf β>0 sup x∈R,y≥0 x + |m|y -β ln E[e xτ +y|W | ] . So for k ≥ K, and i ≥ I k , z k i is in {J |.| ≤ C(z 0 ) + δ}.
By the exact same argument than for J in the proof of Lemma 2.9.4, {J |.| ≤ C(z 0 )+ δ} is a compact level set of J |.| . Then, for k ≥ K and i ≥ I k , z k i is in a compact. There exists at least an adherent point z lim in this compact, and a subsequence z

k j i j → k j ,i j →∞ z lim . In particular, z lim ≥ z 0 .
Like in the proof of Lemma 2.9.5, J n k j (z k j i j ) is given by the inmum for β ∈ (0, β τ ). If β ∈ (0, β τ ) and ε > 0, we may found (x ε , y ε ) such that

βΛ * (1/β, z lim /β) ≤ x ε + my ε -β ln E e xετ +yεW + ε ≤ x ε + z k j i j y ε -β ln E e xετ +yεW + ε + |z lim -z k j i j ||y ε | Since E e xετ +yεW n → E e xετ +yεW
as n growths to innity, for n large enough the dierence is less than ε so that

βΛ * (1/β, z lim /β) ≤ x ε + z k j i j y ε -β ln E e xετ +yεW n + (1 + β τ )ε + |z lim -z k j i j ||y ε | ≤ βΛ * n (1/β, z k j i j /β) + (1 + β τ )ε + |z lim -z k j i j ||y ε |
Taking the inmum in β and then the lim j we get

J(z lim ) ≤ lim inf j J n k j (z k j i j ) + (1 + β τ )ε ≤ lim sup n→∞ inf z≥z 0 J n (z) + (1 + β τ )ε.
By letting ε go to 0 we obtain:

J(z lim ) ≤ lim sup n→∞ inf z≥z 0 J n (z).
Then,

inf z≥z 0 J(z) ≤ J(z lim ) ≤ lim sup n→∞ inf z≥z 0 J n (z).
2.A Lemmata and Propositions from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] The following Lemmata and Propositions are from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]. Some proofs are exactly the same, and we give them to have the totality of the arguments in this manuscript.

Some are slightly dierent from the original article, since we have a third coordinate:

it will be indicated in the title of the Lemmata and Propositions. We can have developed some of the proofs.

2.A.1 Complement to Section 2.5 The functional I n

The following lemma is a rewriting of π(1/τ )H(π|ψ n ) which is useful to work on the rate function I n . Lemma 2.A.1 (Lemma 2.2 from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]). For all π ∈ M 1 ((0, +∞) × {1, ..., n}) such that π(1/τ ) < ∞ and a > 0

π(1/τ )H(π|ψ n ) = sup ϕ∈C b ((0,+∞)×{1,...,n}) (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ )) = sup ϕ,ψ n (e ϕ )=a (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ ) = sup f ∈C((0,+∞)×{1,...,n}),π(f )<∞ (π(f ) -π(1/τ ) ln ψ n (e τ f )). (2.A.1)
Proof. We know that H(π|ψ n ) = sup ϕ∈C b ((0,+∞)×{1,...,n})

(π(ϕ) -ln ψ n (e ϕ )).

Let ϕ such a function, and a = ψ n (e ϕ ). Suppose a > 0, set ϕ a := ϕ -ln a. Then

π(1/τ ) × (π(ϕ) -ln ψ n (e ϕ )) = π(1/τ ) × π (ϕ/τ ) π(1/τ ) -π(1/τ ) ln ψ n (e ϕ ) = π (ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ ) = π (ϕ a /τ ) + ln(a)π (1/τ ) -π(1/τ ) ln (ψ n (e ϕa ) × a) = π (ϕ a /τ ) -π(1/τ ) ln ψ n (e ϕa ).
Moreover ψ n (e ϕa ) = ψ n (e ϕ ) × a -1 = 1. Then the quantity:

sup ϕ,ψ n (e ϕ )=a (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ )) = sup ϕ,ψ n (e ϕ )=1 (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ ))
and doesn't depends on a > 0. Therefore

π(1/τ )H(π|ψ n ) = sup ϕ∈C b ((0,+∞)×{1,...,n}) (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ )) = sup a sup ϕ,ψ n (e ϕ )=a (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ ) = sup ϕ,ψ n (e ϕ )=a (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ ) .
For the last equality, we consider ϕ = τ f . First, for a given ϕ ∈ C b ((0, +∞) × {1, ..., n}), f (τ, W ) := ϕ(τ, W )

τ satises π(f ) < ∞ because: π(f ) = π ϕ τ ≤ ϕ ∞ π(1/τ ) < ∞.
Then:

sup (π(ϕ/τ ) -π(1/τ ) ln ψ n (e ϕ ))
ϕ∈C b ((0,+∞)×{1,...,n})

≤ sup f ∈C((0,+∞)×{1,...,n}) π(f )<∞ (π(f ) -π(1/τ ) ln ψ n (e τ f )).
Secondly, we can restrict the supremum on f to functions such that ψ n (e τ f ) < ∞.

We consider χ M ∈ C b ((0, +∞) × {1, ..., n}) such that χ M ∈ [0, 1], χ M (τ ) = 0 for τ < M -1 and τ > 2M and χ M (τ ) = 1 for τ ∈ [2M -1 , M ].
Then, for f ∈ C((0, +∞) × {1, ..., n}) such that π(f ) < +∞ and ψ n (e τ f ) < +∞,

let ϕ M = τ f χ M . ϕ M is in C b ((0, +∞) × {1, ..., n})
. By dominated convergence, we have:

lim M →∞ π(ϕ M /τ ) -π(1/τ ) ln ψ n (e ϕ M ) = π(f ) -π(1/τ ) ln ψ n (e τ f ).
The equality is then established. Proof. By the lemma 2.A.1, π → π(1/τ )H(π|ψ n ) can be expressed as a supremum of linear functions of π, then this map is convex.

To prove the convexity of I n , let µ 1 and µ 2 in ∆ n , β ∈ (0, 1) and µ = βµ 1 +(1-β)µ 2 . We want to prove

I n (µ) ≤ βI n (µ 1 ) + (1 -β)I n (µ 2 ).
This inequality is true if µ 1 or µ 2 aren't in ∆ n . Let suppose µ is in ∆ n . We denote by α 1 , α 2 , π 1 , π 2 , η 1 and η 2 the corresponding quantities of the equation 2.4.2 for µ 1 and µ 2 .

If

α 1 = α 2 = 0, then µ = βµ 1 + (1 -β)µ 2 = δ (+∞,+∞) ⊗ (βη 1 + (1 -β)η 2 ), I n (µ) = ξ = βξ + (1 -β)
ξ and the inequality is satised.

Else, we have for µ:

α = βα 1 + (1 -β)α 2 π = βα 1 α π 1 + (1 -β)α 2 α π 2 η = β(1 -α 1 ) 1 -α η 1 + (1 -β)(1 -α 2 ) 1 -α η 2 . We notice that 1-α = (β +(1-β))-(βα 1 +(1-β)α 2 ) = β(1-α 1 )+(1-β)(1-α 2 ).
Therefore, by the convexity of π → π(1/τ )H(π|ψ n ),

I n (µ) = απ(1/τ )H(π|ψ n ) + (1 -α)ξ ≤ α βα 1 α π 1 (1/τ )H( π1 |ψ n ) + (1 -β)α 2 α π 2 (1/τ )H( π2 |ψ n ) + (1 -α)ξ ≤ βα 1 π 1 (1/τ )H( π1 |ψ n ) + (1 -β)α 2 π 2 (1/τ )H( π2 |ψ n ) + ξ (β(1 -α 1 ) + (1 -β)(1 -α 2 )) ≤ βI n (µ 1 ) + (1 -β)I n (µ 2 ).
Then I n is convex.

Lemma 2.A.3 (Lemma 2.4 from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]). Let π k ∈ M 1 ((0, +∞) × {1, ..., n}) be such

that π k (1/τ ) < ∞ such that lim k π k (dτ, dW ) = βπ(dτ, dW ) + (1 -β)δ (+∞) (dτ ) ⊗ η 0 (dW ) (2.A.2) for some β ∈ [0, 1], π ∈ M 1 ((0, +∞) × {1, ..., n}) such that π(1/τ ) < ∞ and η 0 ∈ M 1 ({1, ..., n}). Then lim inf k→∞ π k (1/τ )H(π k |ψ k ) ≥ βπ(1/τ )H(π|ψ n ) + (1 -β)ξ.
Proof. By the lemma 2.A.1

π k (1/τ )H(π k |ψ k ) = sup f ∈C((0,+∞)×{1,...,n}), π k (f )<∞ π k (f ) -π k (1/τ ) ln ψ k (e τ f ) . (2.A.3)
To prove this lemma, we dene an adequate function f from some function ϕ. By some optimizations over ϕ and by the equations (2.A.1) from the lemma 2.A.1 we conclude. We x ϕ ∈ C b ((0, +∞) × {1, ..., n}) such that ψ(e ϕ ) < 1. If ξ = 0, we consider d = 0, else we consider d ∈ [0, ξ). For M > 0, let χ M be a smooth function on (0, +∞) such that

χ M (τ ) = 1 for τ ≤ 1 M + 1 or τ ≥ M + 1 χ M (τ ) = 0 for τ ∈ 1 M , M .

Chapter 2. LDP for cumulative processes

We have

ψ n (e dτ χ M +ϕ(1-χ M ) ) = (0, 1 M )×{1,...,n} e dτ χ M (τ )+ϕ(τ,W )(1-χ M (τ )) ψ n (dτ, dW ) + [ 1 M ,M ]×{1,...,n} e ϕ(τ,W ) ψ n (dτ, dW ) + (M +1,+∞)×{1,...,n} e dτ χ M (τ )+ϕ(τ,W )(1-χ M (τ )) ψ n (dτ, dW ) ≤ e ϕ ∞ ψ n e dτ 1 τ < 1 M -→ M →∞ 0 +ψ n (e ϕ ) + e ϕ ∞ ψ n (e dτ 1 τ >M ) -→ M →∞ 0 -→ M →∞ ψ n (e ϕ ) < 1.
Then there exists M which depends on ϕ and d such that ∀M > M , ψ n (e dτ χ M +ϕ(1-χ M ) ) < 1. (The fact that ψ n (e dτ ) is nite implies the convergence of ψ n (e dτ 1 τ >M ) and

ψ n e dτ 1 τ < 1 M to 0.) Since ψ k ψ n , ξ k → ξ > c, we have for k large enough
(depending on M , ϕ and d):

∀M > M , ∀k large enough, ψ k (e dτ χ M +ϕ(1-χ M ) ) < 1. Let f be of the form f (τ, W ) = dχ M (τ ) + ϕ(τ,W ) τ (1 -χ M (τ ))
π k (1/τ )H(π k |ψ k ) ≥ π k (f ) -π k (1/τ ) ln ψ k (e τ f ) ≥ π k (dχ M (τ )) + π k ϕ(τ, W ) τ (1 -χ M (τ )) -π k (1/τ ) ln ψ k (e τ f ) >0 ≥ π k (dχ M (τ )) + π k ϕ(τ, W ) τ (1 -χ M (τ )) .
Then, by (2.5.2) the statement of the lemma lim inf

k→∞ π k (1/τ )H(π k |ψ k ) ≥ lim inf k→∞ π k (dχ M (τ )) + π k ϕ(τ, W ) τ (1 -χ M (τ )) ≥ βπ(dχ M (τ )) + (1 -β)dχ M (+∞) + βπ ϕ(τ, W ) τ (1 -χ M (τ )) + (1 -β) × 0 ≥ βdπ(χ M (τ )) + (1 -β)d + βπ ϕ(τ, W ) τ (1 -χ M (τ ))
We take the limit when M tends to +∞, χ M → 0 point-wisely. Since π(1/τ ) < ∞ and π(τ = +∞) = 0, we obtain by dominated convergence:

βdπ(χ M (τ )) + (1 -β)d + βπ ϕ(τ, W ) τ (1 -χ M (τ )) -→ M →+∞ (1 -β)d + βπ ϕ(τ, W ) τ lim inf k→∞ π k (1/τ )H(π k |ψ k ) ≥ (1 -β)d + βπ ϕ(τ, W ) τ .
We optimize over d < ξ and ϕ such that ψ n (e ϕ ) < 1:

lim inf k→∞ π k (1/τ )H(π k |ψ k ) ≥ sup d<ξ sup ϕ,ψ n (e ϕ )<1 (1 -β)d + βπ ϕ(τ, W ) τ ≥ (1 -β)ξ + β sup ϕ,ψ n (e ϕ )<1 π ϕ(τ, W ) τ .
By the lemma 2.A.1:

sup ϕ,ψ n (e ϕ )<1 π ϕ(τ, W ) τ = sup a<1 sup ϕ,ψ n (e ϕ )=a π ϕ(τ, W ) τ = sup a<1 sup ϕ,ψ n (e ϕ )=a π ϕ(τ, W ) τ -ln ψ n (e ϕ ) + ln ψ n (e ϕ ) = sup a<1 sup ϕ,ψ n (e ϕ )=a π ϕ(τ, W ) τ -ln ψ n (e ϕ ) + ln a = sup a<1 [π(1/τ )H(π|ψ n ) + ln a] = sup a<1 ln a + π(1/τ )H(π|ψ n ) = π(1/τ )H(π|ψ n ).
Then we obtain the announced inequality:

lim inf k→∞ π k (1/τ )H(π k |ψ k ) ≥ βπ(1/τ )H(π|ψ n ) + (1 -β)ξ.
2.A.2 Complement to Section 2.6 Some additional technical topological lemmata.

Lemma 2.A.4 (Decomposition of µ n t (f ) -Equation (3.3) from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]). For f : (0, +∞] 2 × {1, ..., n}) → R, bounded or positive, we have:

µ n t (f ) = 1 t Mt i=1 τ i 1 0 f (uτ i , (1 -u)τ i , W n i )du + τ Mt+1 t t-S M t τ M t +1 0 f (τ Mt+1 u, (1 -u)τ Mt+1 , W n Mt+1 )du (2.A.4)
which can be expressed by:

µ n t (f ) = 1 t Mt i=1 τ i f (1, τ i , W n i ) + τ Mt+1 t f t -S Mt τ Mt+1 , τ Mt+1 , W n Mt+1 .
(2.A.5)

Proof. We have:

µ n t (f ) = 1 t Mt i=1 S i S i-1 f (s -S Ms , S Ms+1 -s, W n Ms+1 )ds + 1 t t S M t f (s -S Ms , S Ms+1 -s, W n Ms+1 )ds = 1 t Mt i=1 1 0 f (uτ i , (1 -u)τ i , W n i )τ i du + 1 t t-S M t τ M t +1 0 f (τ Mt+1 u, (1 -u)τ Mt+1 , W n Mt+1 )τ Mt+1 du, with u = s -S i-1 τ i for each i.
We recall (2.4.6): for all x, τ , W , f (x, τ, w) =

x 0 f (uτ, (1 -u)τ, w)du. We have:

µ n t (f ) = 1 t Mt i=1 τ i f (1, τ i , W n i ) + τ Mt+1 t f t -S Mt τ Mt+1 , τ Mt+1 , W n Mt+1 .
Lemma 2.A.5 (Lemma 2.6 from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]). For all µ ∈ ∆ n , I n (µ) ≤ sup f ∈Γ µ(f ).

Proof. The objective of this proof is to work with a set of functions f , such that f is in Γ and such that µ(f ) is easily calculable. Then, by optimizing over the parameters of f , we prove the wished inequality.

Let M > 0, γ < 1 and ϕ ∈ C c ((0, +∞] × {1, ..., n}) such that ψ n (e ϕ ) = γ.

If ξ = 0, let d = 0, else let d ∈ (0, ξ). Let f d,ϕ,M (a, b, c) = ϕ(a + b, c) a + b + d1 (M,+∞] (a + b), (a, b, c) ∈ (0, +∞] 2 × {1, ..., n}.
First, we prove that f d,ϕ,M is in Γ. f d,ϕ,M is lower semicontinuous on (0, +∞] 2 × {1, ..., n}. In fact, g :

x → g(x 0 ) is lower semicontinuous in x 0 if lim inf x→x 0 g(x) ≥ g(x). On {(a, b, c), a+b < M }, f d,ϕ,M (a, b, c) = ϕ(a+b,c) a+b which is a continuous function (so lower semicontinuous). On {(a, b, c), a + b > M }, f d,ϕ,M (a, b, c) = ϕ(a+b,c) a+b + d which is a continuous function, so lower semicontinuous. Let (a 0 , b 0 , c 0 ) ∈ (0, +∞] 2 × {1, ..., n} such that a 0 + b 0 = M , and let (a k , b k , c k ) → (a 0 , b 0 , c 0 ). Then f d,ϕ,M (a k , b k , c k ) = ϕ(a k + b k , c k ) a k + b k + d1 (M,+∞] (a k + b k ) ≥ ϕ(a k + b k , c k ) a k + b k (because d ≥ 0) lim inf k→∞ f d,ϕ,M (a k , b k , c k ) ≥ ϕ(a 0 + b 0 , c 0 ) a 0 + b 0 = f d,ϕ,M (a 0 , b 0 , c 0 ).

2.A. Lemmata and Propositions from [52]

85

We can study fd,ϕ,M , dened in (2.4.6), for r ∈ [0, 1] and τ > 0:

fd,ϕ,M (r, τ, c) = r 0 f d,ϕ,M (uτ, (1 -u)τ, c)du = r 0 ϕ(τ, c) τ + d1 (M,+∞] (τ )du = r × ϕ(τ, c) τ + d1 (M,+∞] (τ ) .
Then, let s > 0 and

[s,+∞) e τ fd,ϕ,M (s/τ,τ,W ) ψ n (dτ, dW ) = [s,+∞) e τ s/τ × ϕ(τ,W ) τ +d1 (M,+∞] (τ ) ψ n (dτ, dW ) = [s,+∞) e s τ (ϕ(τ,W )+dτ 1 (M,+∞] (τ )) ψ n (dτ, dW ) ≤ [s,+∞) e 1×( ϕ ∞+dτ 1 (M,+∞] (τ )) ψ n (dτ, dW ) ≤ e ϕ ∞ [s,+∞) e dτ 1 (M,+∞] (τ ) ψ n (dτ, dW ) ≤ e ϕ ∞ ψ n (e dτ ).
This integral is bounded uniformly in s, so

D n,f d,ϕ,M = sup s>0 (s,+∞)
e τ fd,ϕ,M (s/τ,τ,W ) ψ n (dτ, dW ) < ∞.

Since ψ n (e ϕ ) = γ < 1, then there exists M 0 depending on c and ϕ such that

∀M > M 0 , C n,f d,ϕ,M = ψ n (e τ fd,ϕ,M (1,τ,W ) ) = ψ n (e ϕ(τ,W )+dτ 1 (M,+∞] (τ ) ) = ψ n (e ϕ(τ,W ) 1 τ ≤M + e dτ +ϕ(τ,W ) 1 τ >M ) ≤ ψ n (e ϕ ) + e ϕ ∞ ψ n (e dτ 1 τ >M ) < 1. Therefore, f d,ϕ,M ∈ Γ. Let µ ∈ ∆ n , µ = αµ 0 + (1 -α)δ (+∞,+∞) ⊗ η. Since ϕ has a compact support: µ(f d,ϕ,M ) = αµ 0 (f d,ϕ,M ) + (1 -α)(δ (+∞,+∞) ⊗ η)(f d,ϕ,M ) = α [0,1]×(0,+∞)×{1,...,n} f d,ϕ,M (uτ, (1 -u)τ, W )(da, db, dc)du ⊗ π(dτ, dW ) + (1 -α)d = α [0,1]×(0,+∞)×{1,...,n} ϕ(τ, W ) τ + d1 (M,+∞] (τ )du ⊗ π(dτ, dW ) + (1 -α)d = απ ϕ(τ, W ) τ + αdπ(τ > M ) + (1 -α)d ≥ απ ϕ(τ, W ) τ + (1 -α)d.

Chapter 2. LDP for cumulative processes

We know that π(ϕ/τ ) = π(1/τ )π(ϕ), then by lemma 2.A.1,

sup f ∈Γ µ(f ) ≥ sup γ<1 sup ϕ,ψ n (e ϕ )=γ sup d<ξ sup M >M 0 µ(f d,ϕ,M ) ≥ sup γ<1 sup ϕ,ψ n (e ϕ )=γ sup d sup M απ ϕ(τ, W ) τ + (1 -α)d ≥ sup γ<1 sup ϕ,ψ n (e ϕ )=γ sup d απ ϕ(τ, W ) τ + (1 -α) ≥ α sup γ<1 sup ϕ,ψ n (e ϕ )=γ π ϕ(τ, W ) τ + (1 -α)ξ ≥ α sup γ<1 sup ϕ,ψ n (e ϕ )=γ π 1 τ [π(ϕ(τ, W )) -ln ψ n (e ϕ ) + ln γ] + (1 -α)ξ ≥ α sup γ<1 sup ϕ,ψ n (e ϕ )=γ π 1 τ [π(ϕ(τ, W )) -ln ψ n (e ϕ )] + π 1 τ ln γ + (1 -α)ξ ≥ απ 1 τ H(ϕ|ψ n ) + sup γ<1 π 1 τ ln γ + (1 -α)ξ ≥ απ 1 τ H(ϕ|ψ n ) + (1 -α)ξ = I n (µ) Then, we have ∀µ ∈ ∆ n , I n (µ) ≤ sup f ∈Γ µ(f ).
Lemma 2.A.6 (Lemma 2.7 from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]). For all µ ∈ ∆n \∆ n , sup f ∈Γ µ(f ) = +∞. Proof. To prove this equality, we determine for each µ ∈ ∆n \∆ n an adequate set of functions f ε , such that f ε is in Γ and such that µ(f ε ) tends to the innity when ε tends to 0.

Let µ ∈ ∆n \∆ n : there exist corresponding α, η and π. Since µ / ∈ ∆ n , α > 0 and π(1/τ ) = +∞. Fix constants δ, d, D and M > 0 such that:

ψ n (τ ∈ (0, δ])(e d -1) < ψ n (τ ∈ (δ, M ])(1 -e -D ).
Such constants exist because this inequality is veried when d and δ → 0, and M and D → ∞. For ε ∈ (0, δ), let f ε be dened by: e τ fε(s/τ,τ,W ) ψ n (dτ, dW ) = (s,+∞)

f ε (a, b, c) = d a + b 1 (ε,δ] (a + b) - D a + b 1 (δ,M ] (a + b). First, we prove that f ε is in Γ. fε (x, τ, W ) = x 0 f ε (uτ, (1 -u)τ, W )du = x 0 d τ 1 (ε,δ] (τ ) - D τ 1 (δ,M ] (τ )du = x d τ 1 (ε,δ] (τ ) - D τ 1 (δ,M ] (τ ) .
e τ s/τ ( d τ 1 (ε,δ] (τ )-D τ 1 (δ,M ] (τ )) ψ n (dτ, dW ) = (s,+∞) e s τ (d1 (ε,δ] (τ )-D1 (δ,M ] (τ )) ψ n (dτ, dW ) ≤ (s,+∞) e s τ (d1 (ε,δ] (τ )) ψ n (dτ, dW ) ≤ (s,+∞) e d ψ n (dτ, dW ) ≤ e d , so D n,fε < +∞. Then f ε ∈ Γ.
On the other hand, Proof. Since C n,f ∈ (0, +∞), we can introduce the probability measure ψ n f (dτ, dV ) = 1/C n,f × e τ f (1,τ,V ) ψ n (dτ, dV ). We denote by ζ k the law of S k if (τ i ) i is i.i.d with 88 Chapter 2. LDP for cumulative processes common law ψ n f .

µ(f ε ) = απ d τ 1 (ε,δ] (τ ) - D τ 1 (δ,M ] (τ ) = αdπ 1 τ 1 (ε,δ] (τ ) -αDπ 1 τ 1 (δ,M ] (τ ) . Since π(1/τ ) = +∞, π 1 τ 1 τ ∈(0,δ] = +∞ and µ(f ε ) -→ ε→0 +∞. Then ∀µ ∈ ∆n \∆ n , sup f ∈Γ µ(f ) = +∞.

E(e tµ

n t (f ) ) = E exp t 0 f (A s , B s , C n s )ds = E 1 Mt=0 exp τ 1 f t τ 1 , τ 1 , W n 1 + ∞ k=1 E 1 Mt=k exp k i=1 τ i f (1, τ i , W n i ) + τ k+1 f t -S k τ k+1 , τ k+1 , W n k+1 = (t,+∞)×{1,...,n} e τ f (1/τ,τ,V ) ψ n (dτ, dV ) + +∞ k=1 E 1 Mt=k k i=1 e τ i f (1,τ i ,W n i ) exp τ k+1 f t -S k τ k+1 , τ k+1 , W n k+1 = (t,+∞)×{1,...,n} e τ f (1/τ,τ,V ) ψ n (dτ, dV ) + +∞ k=1 [0,t] (t-s,+∞) C k n,f e τ f ((ts/τ,τ,V )) ψ n (dτ, dV )ζ k (ds) ≤ D f + +∞ k=1 C k n,f D n,f = D n,f 1 -C n,f
.

Then sup t>0 E(e tµ n t (f ) ) < +∞.

2.A.3 Complement to

lim M →+∞ lim sup t→+∞ 1 t ln P µ n t 1 a + b > M = -∞.
Proof. Let M > 0. We recall that {S n ≤ t} = {M t + 1 > n} and t-S M t τ M t +1 ≤ 1. If M t ≥ 1, we have:

µ n t 1 a + b > M = M t + t -S Mt τ Mt+1 > M t ⊂ {M t + 1 > M t } ⊂ S M t ≤ t .
2.A. Lemmata and Propositions from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] 89 Therefore, by the Markov inequality:

P µ n t 1 a + b > M ≤ P S M t ≤ t ≤ E e -S M t e t = E e -τ 1 M t e t .
Then, we have:

1 t ln P µ n t 1 a + b > M ≤ 1 t t + M t ln E e -τ 1 = 1 + M t t ln E e -τ 1 -→ t→∞ 1 + M ln E e -τ 1 .
Eventually, we have:

lim M →+∞ lim t→+∞ 1 t ln P µ n t 1 a + b > M = -∞.
We also prove that the empirical measure µ n t is close to ∆ n , and we give the

ν n t (f ) := 1 t Mt i=1 τ i 1 0 f (uτ i , (1 -u)τ i , W n i )du + t -S Mt t f (+∞, +∞, W n Mt+1 ) .
Then ν n t ∈ ∆ n almost surely. For all g, δ there exists some t(g, δ) such that for t ≥ t(g, δ), the events {|µ n t (g)ν n t (g)| > δ} and µ n t 1 a+b -ν n t 1 a+b > δ are almost surely empty.

Proof. We can see that ν n t ∈ ∆ n a.s. with α =

S M t t ∈ [0, 1], π n = 1 S M t Mt i=1 τ i δ (τ i ,W n i ) and η = δ W n M t +1
. If S Mt = 0 (ie M t = 0), then α = 0, so π n can be any probability measure such that π n (1/τ ) < ∞.

Else,

π n 1 τ = 1 S Mt Mt i=1 τ i 1 τ i = M t S Mt < ∞ a.s.

Chapter 2. LDP for cumulative processes

Now, let f ∈ C b ((0, +∞] 2 × {1, ..., n}). We remind the decomposition (2.A.5):

µ n t (f ) = 1 t Mt i=1 τ i f (1, τ i , W n i ) + τ Mt+1 t f t -S Mt τ Mt+1 , τ Mt+1 , W n Mt+1 with f (x, τ, w) = x 0 f (uτ, (1 -u)τ, w)du.
We can rewrite the second term

τ Mt+1 t f t -S Mt τ Mt+1 , τ Mt+1 , W n Mt+1 = t -S Mt t 1 0 f (u(t-S M ), τ Mt+1 -u(t-S M ), W n Mt+1 )du.
Then:

|µ n t (f ) -ν n t (f )| = t -S Mt t 1 0 f (u(t -S Mt ), τ Mt+1 -u(t -S Mt ), W n Mt+1 ) -f (+∞, +∞, W n Mt+1 )du
We have t -S Mt ≤ τ Mt+1 by denition.

We denote ξ(s, k) := We have f ∈ C b ((0, +∞] 2 × {1, ..., n}) so f (a, b, c) → f (+∞, +∞, c) as (a, b) → (+∞, +∞). f is bounded so ξ is well dened and nite. Moreover, ξ is bounded (and E too). For each k, ξ(s, k) tends to 0 as s → +∞. So, for u ∈ (0, 1), For each k ≤ n, ξ(., k) is bounded by a constant M and tends to 0 as s → ∞, so E is bounded by M , tends to 0 as s → ∞ and is monotone non-increasing. [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] 91 Then:

f (us, T -us, k) -→ s→∞,T ≥s f (+∞, +∞, k),

2.A. Lemmata and Propositions from

|µ n t (f ) -ν n t (f )| ≤ t -S Mt t 1 0 f (u(t -S Mt ), τ Mt+1 -u(t -S Mt ), W n Mt+1 ) -f (+∞, +∞, W n Mt+1 ) du ≤ t -S Mt t ξ(t -S Mt , W n Mt+1 ) ≤ t -S Mt t E(t -S Mt ).
Finally

{|µ n t (f ) -ν t (f )| > δ} ⊂ t -S Mt t E(t -S Mt ) > δ = {x t E(x t × t) > δ}, where x t = t-S M t t ∈ [0, 1]. If x ∈ [0, 1] satises xE(tx) > δ, then E(tx) > δ. Then x ≤ E -1 (δ)/t. So that δ < xE(tx) < E -1 (δ)/t × M . When t ≥ M E -1 (δ)/δ, this is impossible. Therefore, for t large enough, the event {|µ n t (f ) -ν n t (f )| > δ} is empty.
The proof for the function (a, b, c) → 1 a+b is already given in the proof following Lemma 2.7.3.

Structure of the Proof of the upper bound

We give all the ingredients of the proof in the Section 2.7, and we refer to [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] for the proof. Here, the structure of this proof:

Study of P n t (O) for an open set O to exhibit a rst rate function: For A measurable subset of M 1 ((0, +∞] 2 × {1, ..., n}) and f ∈ Γ, we have:

1 t ln P n t (A) = 1 t ln E e tµ n t (f ) e -tµ n t (f ) 1 A (µ n t ) ≤ 1 t ln E e tµ n t (f ) e -t inf µ∈A µ(f ) ≤ 1 t ln E e tµ n t (f ) -inf µ∈A µ(f ) ≤ 1 t ln D n,f 1 -C n,f -inf µ∈A µ(f ),
where the last inequality is obtained thanks to the Lemma 2.6.

(O ∩ ∆ n M,g,δ )), -R n M,g,δ ≤ - inf µ∈O∩∆ n M,g,δ µ(f ) ∧ R n M,g,δ .
Recall the denition of I n f,M,g,δ (µ) in (2.7.3), for all g, M , δ and all f ∈ Γ,

I n f,M,g,δ (µ) = µ(f ) ∧ R n M,g,δ if µ ∈ ∆ n M,g,δ +∞ otherwise.
, thus, as announced in Section 2.7 , for all open set O, all g, M , δ and all f ∈ Γ, lim sup

t→+∞ 1 t ln P n t (O) ≤ -inf ν∈O I n f,M,g,δ (µ).
(2.A.7)

Large deviations upper bound on compact sets By Lemma 2.B.2, since f is lower semicontinuous, I n f,M,g,δ (µ) is also lower semicontinuous thanks to the compactness of ∆ n M,g,δ (Lemma 2.7.2). By minimizing (2.A.7) over {f, δ, g, M } we obtain:

lim sup t→+∞ 1 t ln P n t (O) ≤ -sup f,M,g,δ inf µ∈O I n f,M,g,δ (µ).
We apply the minimax lemma with -I n f,M,g,δ [Kipnis Landim, Appendix 2.3, Lemma 2.3, [START_REF] Kipnis | Scaling Limits of Interacting Particle Systems[END_REF]] on this equation and we get for all compact set K lim sup

t→+∞ 1 t ln P n t (O) ≤ -inf µ∈K sup f,M,g,δ I n f,M,g,δ (µ).
So (P n t ) t≥0 satises a large deviations upper bound on compact sets with speed t and rate Ĩ(µ) for µ ∈ M 1 ((0, +∞] 2 × {1, ..., n}) where Ĩ(µ) := sup{I n f,M,g,δ (µ), f ∈ Γ, M > 0, g ∈ C b ((0, +∞] 2 × {1, ..., n}), δ > 0}.

Conclusion: As proved in Section 2.7, we have

Ĩ(µ) ≥ sup f ∈Γ I f (µ),
2.A. Lemmata and Propositions from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] 93 where

I f (µ) = µ(f ) if µ ∈ ∆n +∞ otherwise.
.

According to lemma 2.6.4, Ĩ(µ) ≥ I n (µ).

Then (P n t ) t≥0 satises a large deviations upper bound with rate I n on compact sets. The exponential tightness of (µ t ) t follows the Lemma 2.7.1. By the Lemma 1.2.18 from [Dembo Zeitouni, [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]], (P n t ) t≥0 satises the full large deviations upper bound on closed sets.

2.A.3.2 Lower bound

The proof of the lower bound is very similar to [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF], just replacing the sample τ i by a sample (τ i , W i ).

We prove: Theorem

Chapter 2. LDP for cumulative processes

Then, by using Jensen's inequality:

ln P n t (V) = ln V dP n t dQ t dQ t = ln 1 Q t (V) V dP n t dQ t dQ t + ln Q t (V) ≥ 1 Q t (V) V ln dP n t dQ t dQ t + ln Q t (V) ≥ - 1 Q t (V) V ln dQ t dP n t dQ t dP n t dP n t + ln Q t (V) ≥ 1 Q t (V) -H(Q t |P n t ) + V c ln dQ t dP n t dQ t dP n t dP n t + ln Q t (V).
Moreover, ∀x ≥ 0, x ln x ≥ -e -1 and we obtain:

ln P n t (V) ≥ 1 Q t (V) -H(Q t |P n t ) - V c e -1 dP n t + ln Q t (V) ≥ 1 Q t (V) -H(Q t |P n t ) -e -1 + ln Q t (V). However, Q t δ µ , µ ∈ V which is open, then Q t (V) -→ t→∞ 1. We obtain: lim inf t→+∞ 1 t ln P n t (V) ≥ -lim sup t→∞ 1 t H(Q t |P n t ) ≥ -I n (µ).

Proof of Proposition 2.A.11

To prove this Proposition, we just need a law of large numbers for µ n t . We denote by P π the law of an i.i.d. sequence of (τ i , W i ) i with marginal distribution π.

Proposition 2.A.12 (Proposition 4.1 of the article). Let π ∈ M 1 ((0, +∞]×{1, ..., n})

with π(1/τ ) ∈ (0, +∞). Under P π, a.s,

µ n t t→∞ [0,1]×(0,+∞)×{1,...,n} δ (uτ,(1-u)τ,V ) du ⊗ π(dτ, dV ).
Proof. With the law of the large numbers and the renewal theorem, we show that under P π, for each f ∈ C b ((0, +∞] 2 × {1, ..., n}), µ n t (f ) converges a.s.

Let f ∈ C b ((0, +∞] 2 × {1, ..., n}), we remind (2.A.5),

µ n t (f ) = 1 t Mt i=1 τ i f (1, τ i , W n i ) + τ Mt+1 t f t -S Mt τ Mt+1 , τ Mt+1 , W n Mt+1 .
By the strong law of large numbers, under P π, a.s.

lim n→+∞ 1 n n i=1 τ i f (1, τ i , W n i ) = π(τ f (1, τ, W n )) = 1 π(1/τ ) π( f (1, τ, W n )).
By the renewal theorem, a.s.

lim t→+∞ M t t = 1 E π(τ ) = π(1/τ ) τ 1 τ π(dτ ) = π(1/τ )
Therefore, under P π a.s.

lim t→+∞ 1 t Mt i=1 τ i f (1, τ i , W n i ) = lim t→+∞ M t t 1 M t Mt i=1 τ i f (1, τ i , W n i ) = π( f (1, τ, W n ))
By the law of large numbers, a.s.

lim k→+∞ S k k = π(τ ) = 1 π(1/τ ) , so, a.s, lim t→+∞ S Mt t = lim t→+∞ S Mt M t M t t = 1 and lim t→+∞ t -S Mt t = 0.
It follows that a.s.

lim t→+∞ τ Mt+1 t f t -S Mt τ Mt+1 , τ Mt+1 , W n Mt+1 ≤ lim t→+∞ t -S Mt t f ∞ = 0.
So for all f ∈ C b ((0, +∞] 2 × {1, ..., n}), P π a.s.

µ n t (f ) -→ t→∞ π( f (1, τ, W n )) = [0,1]×(0,+∞)×{1,...,n} f (uτ, (1 -u)τ, V )du ⊗ π(dτ, dV ).
Proof of Proposition 2.A.11. First, we prove this proposition for µ ∈ ∆ n 0 . We con- struct a sequence Q n,δ t of laws, such that lim δ→0 lim t→+∞ Q n,δ t = δ µ . By using the contraction principle on the relative entropy and selecting a good δ(t), we can prove that the expression above is true. Eventually, we extend this proposition to ∆ n , by using the proposition 2.5.6: for each µ ∈ ∆ n , there exists a good sequence µ k ∈ ∆ n 0 .

We suppose µ ∈ ∆ n 0 , with associated π. Notice that µ(1/τ ) = π(1/τ ) ∈ (0, +∞). Fix δ > 0 and set T t,δ = (1 + δ)tπ(1/τ ) . For t > 1 π(1/τ ) , let us denote by P n,δ t the law on ((0, +∞) × {1, ..., n}) N * such that under P n,δ t , the sequence (τ i , W i ) i is independent and:

1. for all i ≤ T t,δ , (τ i , W i ) has law π,

2. for all i ≥ T t,δ + 1, (τ i , W i ) has law ψ.

Chapter 2. LDP for cumulative processes

We set Q n,δ t = P n,δ t • (µ n t ) -1 . We will prove that: lim δ→0 lim t→+∞ Q n,δ t = δ µ . By the law of large numbers of proposition 2.A.12, under P π we have a.s.

lim k→+∞ S k k = E π(τ ) = τ 1 τ π(1/τ ) π(dτ ) = 1 π(1/τ ) . So lim t→+∞ S T t,δ t = lim t→+∞ S T t,δ T t,δ T t,δ t = 1 π(1/τ ) (1 + δ)π(1/τ ) = 1 + δ
Moreover, S T t,δ has the same law under P π and P n,δ t so, for any δ > 0,

lim t→+∞ P n,δ t (S T t,δ ≤ t) = lim t→+∞ P π(S T t,δ ≤ t) = lim t→+∞ P π S T t,δ t ≤ 1 = 0.
We set D t,δ = {S T t,δ > t}. We have for all δ > 0, lim t→+∞ P n,δ t (D t,δ ) = 1. We recall that {S n > t} = {M t + 1 ≤ n}. Therefore, on D t,δ we have M t + 1 ≤ T t,δ and therefore for any f ∈ C b ((0, +∞] 2 × {1, ..., n}) and ε > 0,

P n,δ t (|µ n t (f ) -µ(f )| > ε) ≤ P n,δ t ({|µ n t (f ) -µ(f )| > ε} ∩ D t,δ ) + P n,δ t (D c t,δ )
We have lim t→+∞ P n,δ t (D c t,δ ) = 0. On D t,δ , P n,δ t and P π coincide, so

P n,δ t ({|µ n t (f ) -µ(f )| > ε} ∩ D t,δ ) = P π({|µ n t (f ) -µ(f )| > ε} ∩ D t,δ ) ≤ P π(|µ n t (f ) -µ(f )| > ε) -→ t→+∞ 0,
where we use Proposition 2.A.12.

It implies that, for all δ > 0, lim t→+∞ Q n,δ t = δ µ . In particular, we have:

lim δ→0 lim t→+∞ Q n,δ t = δ µ .
Now we estimate the entropy H(Q n,δ t |P t ), by using the contraction principle

H(P n,δ t • (µ n t ) -1 |P ψ • (µ n t ) -1 ) ≤ H(P n,δ t |P ψ ). H(Q n,δ t |P t ) ≤ H(P n,δ t |P ψ ) = H     i≤T t,δ π  ⊗   i>T t,δ ψ     i≤T t,δ ψ   ⊗   i>T t,δ ψ     = T t,δ H(π|ψ).

2.A. Lemmata and Propositions from [52]

97 So:

lim δ→0 lim sup t→+∞ 1 t H(Q n,δ t |P t ) ≤ lim δ→0 lim sup t→+∞ 1 t T t,δ H(π|ψ) = π(1/τ )H(π|ψ).
Then, there exists a function t → δ(t) > 0 vanishing as t → +∞ such that

Q t = Q n,δ(t) t → δ µ and lim sup t→+∞ 1 t H(Q t |P t ) ≤ I n (µ).
We suppose now µ ∈ ∆ n \∆ n 0 .

By proposition 2.5.6 we can nd a sequence (µ k ) k in ∆ n 0 such that µ k µ and lim sup I n (µ k ) ≤ I n (µ). Moreover, for all k, there exists a family Q t,k on M 1 ((0, +∞] 2 × {1, ..., n}) such that Q t,k δ µ k and:

lim sup

t→+∞ 1 t H(Q t,k |P t ) ≤ I n (µ k )
With a diagonal procedure, we can nd a family Q t such that Q t δ µ and lim sup

t→∞ 1 t H(Q t |P t ) ≤ I n (µ).

2.A.4 Complement to Section 2.8 Proof of Theorem 2.4.4

We recall the notations from Section 2.8.

We dene for ε > 0

ϕ ε (a, b, c) = c (a + b) ∨ ε . Replacing τ i by τ ε i = τ i ∨ ε we may introduce S ε n , M ε t , A ε t , B ε t , C n,ε
t and µ n,ε t as in subsection 2.4.1. We have the following µ ε,n t (ϕ ε ) -

1 t Mt i=1 W n i ≤ n t ((M t -M ε t ) + 1) .
Lemma 2.A.13 (Lemma 5.4 from [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF]). (M ε t /t) t≥0 is an exponentially good approximation of (M t ) t , i.e. for all δ > 0, lim ε→0 lim sup

t→+∞ 1 t ln P (|M t -M ε t | > tδ) = -∞
Proof. Since M ε t ≤ M t , we study for δ > 0 and C > 0

P (M t -M ε t > tδ) ≤ Ct i=0 P (M t -M ε t > tδ, M t = i) + P(M t > Ct) = Ct i=0 i k=0 P   M t -M ε t > tδ, M t = i, i j=1 1 τ j <ε = k   + P(S Ct ≤ t).

Chapter 2. LDP for cumulative processes

For k ≤ i ≤ Ct and m ≤ n, on the event {M t = n, M ε t = m, i j=1 1 τ j <ε = k}, we have:

t < S ε m ≤ S m + kε, then S m > t -
Z n t t > δ = -∞ .
Proof. In fact, we have

µ ε,n t (ϕ ε ) - Z n t t ≤ n t ((M t -M ε t ) + 1) , thus for δ > 0, lim ε→0 lim sup t→∞ 1 t ln P µ ε,n t (ϕ ε ) - Z n t t > δ ≤ lim ε→0 lim sup t→∞ 1 t ln P M t -M ε t > δt n -1 = -∞.
2.B Complementary proofs

H(ν 1 |ν 2 ) ≥ M m=1 ν 1 (B m ) ln ν 1 (B m ) ν 2 (B m ) .
Proof. By the properties of the relative entropy:

H(ν 1 |ν 2 ) = sup f f dν 1 -ln e f dν 2 ,
where the supremum is taken over all f that are measurable and bounded.

For

f = M m=1 ln ν 1 (B m ) ν 2 (B m ) 1 Bm ,
we have:

e M m=1 ln ν 1 (Bm) ν 2 (Bm) 1 Bm dν 2 = M n=1 Bn e M m=1 ln ν 1 (Bm) ν 2 (Bm) 1 Bm dν 2 = M n=1 Bn e ln ν 1 (Bn) ν 2 (Bn) dν 2 = M n=1 ν 1 (B n ) ν 2 (B n ) ν 2 (B n ) = 1 f dν 1 -ln e f dν 2 = M m=1 ln ν 1 (B m ) ν 2 (B m ) 1 Bm dν 1 -ln e M m=1 ln ν 1 (Bm) ν 2 (Bm) 1 Bm dν 2 = M m=1 ln ν 1 (B m ) ν 2 (B m ) ν 1 (B m ) µ ∈ U, Φ(µ) = R > t.
Secondly, if µ 0 ∈ E, then Φ(µ 0 ) = µ 0 (f )∧R. We have already proved that µ → µ(f ) is lower semi-continuous. Then µ → µ(f ) ∧ R is lower semi-continuous. Indeed: if µ 0 (f ) ≤ R, then Φ(µ 0 ) = µ 0 (f ). Then for every t < µ 0 (f ), there exists a neighborhood U of µ 0 such that ∀ν ∈ U, ν(f

) > t. As t < µ 0 (f ) ≤ R, Φ(ν) = ν(f ) ∧ R > t. if µ 0 (f ) > R, then Φ(µ 0 ) = R.
Then for every t < µ 0 (f ), there exists a neighborhood U of µ 0 such that ∀ν ∈ U, ν(f ) > t. In particular, for

t = R, ∀ν ∈ U, ν(f ) > R and Φ(ν) = R. Then for every s < R = Φ(µ 0 ), ∀ν ∈ U, Φ(ν) = R > s. Eventually, if µ 0 ∈ E \ E: let t < Φ(µ 0 ). In particular, R > t. There exists a neighborhood U 1 of µ 0 in E such that: ∀ν ∈ U 1 , Φ(ν) > t. Let U be a neighborhood of µ 0 in M 1 ((0, +∞] 2 ) such that U ∩ E ⊂ U 1 . For all ν ∈ U ∩ E c , Φ(ν) = R > t,
and for all ν ∈ U ∩ E, Φ(ν) > t. So Φ is a lower semi-continuous function.

Chapter 3

Limit theorems for Hawkes processes including inhibition. This chapter consists in the article [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF], accepted for publication in Stochastic Processes and their Applications. Notice that the companion paper [START_REF] Cattiaux | Large Deviation Principles for Cumulative Processes and Applications[END_REF] cited several times is the Chapter 2.

Abstract

In this paper we consider some non linear Hawkes processes with signed reproduction function (or memory kernel) thus exhibiting both self-excitation and inhibition. We provide a Law of Large Numbers, a Central Limit Theorem and large deviation results, as time growths to innity. The proofs lie on a renewal structure for these processes introduced in [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF] which leads to a comparison with cumulative processes. Explicit computations are made on some examples. Similar results have been obtained in the literature for self-exciting Hawkes processes only. Hawkes processes have been introduced by Hawkes [START_REF] Hawkes | Spectra of Some Self-Exciting and Mutually Exciting Point Processes[END_REF] and are widely used for modeling purposes: originally as models for the appearances of earthquakes [START_REF] Hawkes | Spectra of Some Self-Exciting and Mutually Exciting Point Processes[END_REF][START_REF] Hawkes | A Cluster Process Representation of a Self-Exciting Process[END_REF], Chapter 3. Limit theorems for Hawkes processes but now in nance [START_REF] Hawkes | Hawkes Processes and Their Applications to Finance: A Review[END_REF][START_REF] Bacry | Hawkes Model for Price and Trades High-Frequency Dynamics[END_REF] and econometrics or in neuroscience as models of spike trains of neurons [START_REF] Löcherbach | Spiking Neurons : Interacting Hawkes Processes, Mean Field Limits and Oscillations[END_REF][START_REF] Richard Hansen | Lasso and Probabilistic Inequalities for Multivariate Point Processes[END_REF]. We refer to the bibliography of our references for more details.

A Hawkes process t → N h t = N h ([0, t]) is a point process on the real line R characterized by its initial condition on ] -∞, 0] and its intensity process t → Λ(t) through the innitesimal relation

P(N h . has a jump in ]t, t + dt[|F t ) = Λ(t) dt ,
where

F t = σ(N h (] -∞, s[ ; s ≤ t))
is the natural ltration of the process and

Λ(t) = f λ + ]-∞,t[ h(t -s) N h (ds) . (3.1.1)
Here λ ∈ R, f : R → R + is the jump rate function and h : R + → R is the reproduction function (or memory kernel). We shall give a more precise denition in the next section (in particular on what happens before time 0) as well as results on existence and stability.

When f is linear or ane, the process is said to be linear. In this case one has to assume that λ ≥ 0 and h ≥ 0 too. Note that when h vanishes identically we recover a standard Poisson process. Otherwise the Hawkes process is called non linear. Actually, except for the behaviour of the shifted process ( [START_REF] Massoulié | Stability Results for a General Class of Interacting Point Processes Dynamics, and applications11This Work Was Partially Done While the Author Was with the Laboratoire Des Signaux et Systèmes[END_REF][START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF]), very few papers are dealing with possibly negative or signed h. The negative part of h can be interpreted as self-inhibition.

It is very natural to look at the large time behaviour of N h . , in particular the Law of Large Numbers (LLN) the Central Limit Theorem (CLT) and the deviations from the asymptotic mean or more generally the large deviations (LD).

In the linear case (recall that h is thus assumed to be non-negative) and assuming that

h L 1 (du) < 1, both the LLN N h t t → λ 1-h L 1 (du) := µ a.s. as t → +∞ , (3.1.2)
and the CLT

N h t -µt √ t =⇒ N h (0, σ 2 ) with σ 2 = λ (1-h L 1 (du) ) 3 ,
where the convergence holds in distribution, have been shown (see e.g. [START_REF] Daley | An Introduction to the Theory of Point Processes[END_REF]). Actually Bacry et al. [START_REF] Bacry | Some Limit Theorems for Hawkes Processes and Application to Financial Statistics[END_REF] have obtained the functional version of the CLT (convergence to some Brownian motion) in the multivariate case. In a dierent direction, [START_REF] Gao | Functional Central Limit Theorems for Stationary Hawkes Processes and Application to Innite-Server Queues[END_REF] have shown a CLT for xed t as λ → +∞.

The easiest way to derive LLN and CLT in the linear case is presumably to use the immigration-birth representation also called the cluster process representation in [START_REF] Hawkes | A Cluster Process Representation of a Self-Exciting Process[END_REF], connecting N h to subcritical Galton-Watson processes. This representation 3.1. Introduction. 105 was used in [START_REF] Bordenave | Large Deviations of Poisson Cluster Processes[END_REF] in order to get the Large Deviation (LD) principle for N h t /t with rate function

I(x) = x ln x λ + x h L 1 (du) -x(1-h L 1 (du) ) + λ .
For this explicit expression of the rate function see [START_REF] Zhu | Central Limit Theorem for NonLinear Hawkes Processes[END_REF] p.761. The LD principle is obtained in [START_REF] Bordenave | Large Deviations of Poisson Cluster Processes[END_REF] under the additional assumption [START_REF] Gao | Precise Deviations for Hawkes Processes[END_REF] that this assumption is not necessary. Under more restrictive assumptions, [START_REF] Gao | Precise Deviations for Hawkes Processes[END_REF] contains precise deviations (see e.g. Theorem 2 therein).

+∞ 0 t h(t) dt < +∞. It is claimed in the introduction of
The non linear case is of course more dicult. According to the general seminal paper by Brémaud and Massoulié [START_REF] Brémaud | Stability of Nonlinear Hawkes Processes[END_REF], if

f is L-Lipschitz and L h L 1 (du) < 1,
there exists a unique stationary version of the Hawkes process. Rate of convergence to equilibrium is studied in [START_REF] Brémaud | Rate of Convergence to Equilibrium of Marked Hawkes Processes[END_REF] in two specic cases. As a consequence of Brémaud and Massoulié result, we get that

N h t t → µ = E s [N h ([0, 1])] a.s. as t → +∞ , (3.1.3)
where E s denotes the expectation w.r.t. the stationary ergodic distribution.

In the particular situation where h is an exponential, the Hawkes process becomes Markovian and some results of large deviation have been obtained [START_REF] Zhu | Large Deviations for Markovian Nonlinear Hawkes Processes[END_REF]. In [START_REF] Zhu | Central Limit Theorem for NonLinear Hawkes Processes[END_REF], Zhu proved a functional CLT at equilibrium from which the following follows Theorem 3.1.1. Assume that (1)

f is L-Lipschitz, (2)
h is non-negative, decreasing and such that

+∞ 0 t h(t) dt < +∞, (3)
L +∞ 0 h(t) dt < 1, (4)
λ ≥ 0.

Then the stationary Hawkes process satises

N h t -µt √ t =⇒ N h (0, σ 2 ) as t → +∞ in distribution, with σ 2 := Var s (N h ([0, 1])) + 2 j≥1 Cov s (N h ([0, 1]), N h ([j, j + 1]))
where Var s and Cov s denote the variance and covariance w.r.t. the stationary distribution.

The proof is based on martingales techniques for the functional CLT. As the author himself is saying, to obtain an explicit expression for µ and σ 2 can rapidly become a dicult task. In the same work, Zhu also obtained a Strassen iterated logarithm law. One can also mention [START_REF] Zhu | Process-Level Large Deviations for Nonlinear Hawkes Point Processes[END_REF] where a large deviation result is obtained by contracting the level-3 LDP, i.e. by considering the shifted occupation measure.

Theorem 2 in [START_REF] Zhu | Process-Level Large Deviations for Nonlinear Hawkes Point Processes[END_REF] then furnishes a LDP for N h t /t, provided h is non decreasing and non-negative and f is sub-linear at innity. The expression of the rate function, as 106 Chapter 3. Limit theorems for Hawkes processes the inmum of the entropy on some set of measures satisfying a linear constraint is however not really tractable.

Since we are interested in neurosciences, our goal in this work is to understand the role of self-inhibition in the asymptotic behaviour of Hawkes processes. Since inhibition will slow down the neuronal activity, we thus have to consider signed functions h (the positive part modeling the self-excitation), but also jump rate functions f satisfying f (u) = 0 if u ≤ 0. In the present paper, we will study the case of a general, signed, reproduction function with compact support and the specic jump rate function f (u) = u + = max(u, 0). This choice is of course the simplest one allowing us to introduce inhibition, and to compare this situation with linear models.

We will obtain a LLN, a CLT and deviation inequalities, where the parameters are characterized by the renewal structure of the process introduced in [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF] replacing the classical cluster representation of the self-exciting case established in [START_REF] Hawkes | A Cluster Process Representation of a Self-Exciting Process[END_REF] which is no more valid. This renewal structure allows us to write the Hawkes process almost as a cumulative process. The main tools are then limit theorems for cumulative processes and actually, the technical work consists in showing that one can apply these theorems in the present situation. An important tool is a comparison between the considered Hawkes process, the self excited process associated to the positive part of the reproduction function, furnishing an upper bound, and a purely inhibited process corresponding to the (negative) lower bound of the reproduction function (see Proposition 3.2.4), furnishing a lower bound.

For simplicity we restrict ourselves to an empty initial condition (see below). Some explicit computations are done in simple particular cases of pure inhibition (h non-positive). Precise statements will require some denitions, so that they are postponed to the next section. We emphasize, that the inhibition part introduces new intricacies.

As we said, very few papers are dealing with inhibition. In [START_REF] Duarte | Stability, Convergence to Equilibrium and Simulation of Non-Linear Hawkes Processes with Memory Kernels given by the Sum of Erlang Kernels[END_REF] some specic kernels are considered, but the addressed problem is not the one we are considering here. Looking at possibly negative reproduction functions is not only of mathematical interest. As shown in [START_REF] Richard Hansen | Lasso and Probabilistic Inequalities for Multivariate Point Processes[END_REF][START_REF] Reynaud-Bouret | Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis[END_REF][START_REF] Chevallier | Microscopic Approach of a Time Elapsed Neural Model[END_REF] a multivalued version of the model we are studying is particularly well suited for modeling spike train of neurons, at least in an almost stationary regime. To extend our results to the multivalued framework should thus be an interesting question.

3.2 Notation, denitions and results.

Hawkes processes

We consider an appropriate ltered probability space (Ω, F, (F t ) t≥0 , P) satisfying the usual assumptions. Let N 0 a locally nite point process on (-∞, 0] with law m. The point process N h on R is a Hawkes process on (0, +∞), with initial condition N 0 and reproduction measure µ(dt) = h(t)dt if:

N h | (-∞,0] = N 0 , the conditional intensity measure of N h | (0,+∞) with respect to (F t ) t≥0 is absolutely continuous w.r.t the Lebesgue measure and has stochastic intensity:

Λ h : t ∈ (0, +∞) → λ + (-∞,t) h(t -u)N h (du) + . (3.2.1)
where x + = max(x, 0).

The next proposition gives an explicit representation of the Hawkes process as solution of an SDE driven by a Poisson point process and states an important coupling property.

Proposition 3.2.2 (Proposition 2.1 in [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF]). Let Q be a (F t ) t≥0 -two-dimensional

Poisson point process on (0, +∞) × (0, +∞) with unit intensity. We consider the equation

   N h = N 0 + (0,+∞)×(0,+∞) δ u 1 θ≤Λ h (u) Q(du, dθ) Λ h (u) = λ + (-∞,u) h(u -s)N h (ds) + , u > 0, (3.2.2)
where λ > 0 is an immigration rate, h : (0, +∞) → R is a signed measurable function and N 0 is an initial condition of law m on (-∞, 0]. We consider the similar equation for N h + in which h is replaced by h + (.) = max(h(.), 0). We assume that h + 1 := h + L 1 (du) < 1 and that the distribution m satises:

∀t > 0, t 0 E m (-∞,0] h + (u -s)N 0 (ds) du < +∞. ( 3

.2.3)

Then:

There exists a pathwise strong solution N h of equation (3.2.2), and this solution is a Hawkes process.

This property is true for N h + . Moreover, in the sense of measures,

N h ≤ N h + , meaning that for all 0 ≤ s ≤ t < +∞, N h ([s, t]) ≤ N h + ([s, t]).

Denitions and assumptions

In this paper we consider a Hawkes process N h according to Denition 3.2.1. We focus on the case of a signed reproduction function h which represents a possible inhibition on the appearance of future points. Assumption 3.2.3. In all the paper, we will make the following assumptions : i) h : (0, +∞) → R is a compactly supported signed measurable function. We dene L(h) as the supremum of the support of h: L(h

) := sup{t > 0, |h(t)| > 0} < ∞. ii) h + 1 := +∞ 0 h + (u) du < 1,
where h + (x) = max(h(x), 0). iii) λ > 0, iv) the initial condition on ] -∞, 0[ does not contain any point i.e. m = δ ∅ .

We are interested in the asymptotic behaviour of the number of jumps of the process N h on the interval [0, t], and we denote:

N h t = N h ([0, t]), ∀t ≥ 0
In particular we aim at quantifying precisely the loss of points due to inhibition.

We will prove asymptotic results for N h t t and give exact computations on specic examples.

First we show another comparison result, this time furnishing a lower bound for N h t . This result motivates the detailed study of the canceling of intensity example.

Proposition 3.2.4 (Minoration of Hawkes process).

Let h be a function satisfying Assumptions 3.2.3. Let λ > 0 and dene g = -λ1 [0,L(h)] .

One can nd a coupling of two Hawkes processes N h and N g , respectively associated with the reproduction functions h and g and with basal intensity λ, such that for any t ≥ 0:

N h t ≥ N g t a.
s.

Note that this comparison result is weaker than the majoration via h + , since we do not have N h ([s, t]) ≥ N g ([s, t]) for all s, but only for s = 0.

Proof. The main idea is to construct these two processes with the same Poisson point process Q on (0, +∞) 2 . We consider the successive jumps of N h : U h 1 , U h 2 , U h 3 , ...; and the ones of N g : U g 1 , U g 2 , U g 3 , ....

We will prove by induction, that

∀j ≥ 1, N h U g j ≥ N g U g j = j a.s.
by studying the intervals associated with the [U g j , U g j+1 ) for j ∈ N. We stress out that considering the denition of the function g, the intensity Λ g of the Hawkes process N g can only take the two values 0 and λ. First interval: First remark that ∀t < min(U h 1 , U g 1 ),

Λ h (t) = λ = Λ g (t),

Notation, denitions and results

.

109

thus we have U h 1 = U g 1 and consequently N h U g 1 = N g U g 1 .
Second interval: For j = 2: by denition, there is only one jump for N g on [U g 1 , U g 2 ). There are two possibilities for N h :

Assume that there is no other jump that U h

1 in this interval. Since U g 2 ≥ U g 1 + L(h), we have Λ g (U g 2 -) = λ = Λ h (U g 2 -). Accordingly, U h 2 = U g 2 and in particular, N h U g 2 = N g U g 2 a.s.
Otherwise, there is at least one other jump of

N h in (U g 1 , U g 1 + L(h)). In this case, N h U g 2 ≥ 2 = N g U g 2 a.s.
Recursion step: We x j and we suppose that the statement holds for i ≤ j. Let k = N h U g j ≥ j by assumption. Then consider the two following cases:

If U g j is a jump of N h ,
Λ h (U g j+1 -) = Λ g (U g j+1 -) = λ, since U g j+1 > U g j + L(h). So, U h k+1 = U g j+1 . In both situations, N h U g j+1 ≥ 1 + N h U g j ≥ 1 + N g U g j = N g U g j+1 . If U g j is not a jump of N h , then Λ h (U g j -) < λ = Λ g (U g j -).
Therefore since the support of h is of length L(h) we deduce that

U h k < U g j < U h k + L(h).
By the induction hypothesis, we know that k ≥ j. Then, there is either at least one jump of N h in (U g j , U g j + L(h)), or the next jump is U g j+1 , i.e.

U h k+1 = U g j+1 . In both cases, we have N h

U g j+1 ≥ 1 + k ≥ 1 + j = N g U g j+1 .
This concludes the induction. Let us come back to a general t ∈ R + . For any xed ω, there exists j = j(ω) ∈ N, such that: U g j (ω) ≤ t < U g j+1 (ω). Then using the monotonicity of N h we have

N h t ≥ N h U g j ≥ N g U g j = N g t .
Both comparison results may be used in the sequel.

Hawkes processes as cumulative processes.

Our study of the large time behaviour of Hawkes processes lies on a renewal structure for Hawkes processes rst introduced in [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF] we shall partly recall below.

Notice that this structure is used in [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF] for a completely dierent purpose.

(a) Example of function h: a square wave

Λ h (t) U 1 U 2 U 3 U 4 (b)
Example of simulation of a Hawkes process with parameter h: in blue, the intensity function t → Λ h (t); in red, the jumps times. The axis below indicates the Dirac measures of the process.

Figure 3.1 Example of Hawkes process

Let N h be a Hawkes process according to Denition 3.2.1, with initial condition N 0 = ∅. We denote by U 1 , U 2 , U 3 , ... its successive jumps.

Let us introduce the renewal times of the process which splits the time line into independent and identically distributed time windows of length τ 1 , τ 2 , • • • . Dene the stopping time

τ 1 = inf{t > U 1 , N h ((t -L(h), t]) = 0},
that is the rst time after U 1 such that there has been no jump during a time L(h).

We also set S 0 = 0 and S 1 = τ 1 .

Let us now dene

W 1 = N h ([U 1 , S 1 ]) = N h ([0, S 1 ]),
the number of jumps of the process in this rst time window and rename the jump times in the rst time window as:

U 1 j = U j , ∀j ∈ {1, • • • , W 1 }.
We shall see below that τ 1 and W 1 are almost surely nite. Recursively let i ∈ N * such that (τ 1 , W 1 ), ...(τ i , W i ) are well dened (and a.s. nite). Let S i = i k=1 τ k and dene

U i+1 1 = U W 1 +...+W i +1 ,
and

τ i+1 = inf{t > U i+1 1 , N h ((t -L(h), t]) = 0} -S i , (3.2.4)
Notice that there is at least one jump in [S i , S i + τ i+1 ]. We nally introduce the number of jumps in the (i + 1)'th window as

W i+1 = N h ([U i+1 1 , S i + τ i+1 ) = N h ([S i , S i + τ i+1 ]), (3.2.5)

Notation, denitions and results.

111 and rename the associated jump times as:

Λ h (t) U 1 1 ... U 1 4 U 2 1 U 3 1 U 3 2 W 1 = 4 W 2 = 1 W 3 = 2 1 + U 1 4 -U 1 1 1 1 + U 3 2 -U 1 1 0 S 1 S 2 S 3 τ 1 τ 2 τ 3
U i+1 j = U W 1 +...+W i +j , ∀j ∈ {1, ..., W i+1 }.
(t) = -λ1 (1,2) (t), so that L(h) = 2.
The next Proposition gathers important properties on the law of (τ i , W i ) dened above. However more explicit information are dicult to obtain except in specic cases (see Section 3.3).

Proposition 3.2.5. Under Assumptions 3.2.3, and using the above denitions: i) the (τ i , W i ) i are i.i.d. random variables, ii) for i ∈ N * , the (U i 1 -S i-1 ) are i.i.d. random variables with exponential distribution E(λ), that is, the time between the beginning of a window and the rst point of this window follows an exponential law.

Proof. Let Q be a two-dimensional Poisson point process, and let N h generated by Q as in proposition 3.2.2, the (τ i , W i ) i being dened as before.

Given τ 1 , remark that U 2 1 is the rst jump of Q on (τ 1 , +∞) × [0, ∞]. Indeed, using successively the denition of L(h) and τ 1 we deduce that:

Λ(τ 1 ) = λ + (-∞,τ 1 ) h(t -u)N h (du) + = λ + (τ 1 -L(h),τ 1 ) h(t -u)N h (du) + = λ. By translation, U 2 1 -τ 1 is the rst jump of a Poisson point process Q on (0, ∞)× [0, ∞], independent of Q on (0, τ 1 ) × [0, ∞], and U 2 1 -τ 1 is independent of τ 1 = S 1 .
Since the jumps of N h before τ 1 do not inuence Λ h (t) for t > τ 1 (by denition of L(h) and τ 1 ),

τ 2 = inf{t > U 2 1 -S 1 , N h ((t + S 1 -L(h), t + S 1 ]) = 0}, only depends on Q on (U 2 1 , +∞)×[0, ∞]. Moreover, (0, S 1 )×[0, ∞] and (U 2 1 , +∞)× [0, ∞] are almost surely disjoints. Hence Q on (U 2 1 , +∞) × [0, ∞] is independent of Q on (0, S 1 ) × [0, ∞] so that τ 2 is independent of (τ 1 , W 1 ).
The number of points in the second time window

W 2 = N h ([U 2 1 , S 2 ]) = N h ([S 1 , S 2 ]) only depends on Q on (U 2 1 , +∞) × [0, ∞]. W 1 depends on Q on (0, S 1 ) × [0, ∞].
For the same reason as before, W 2 is independent of (τ 1 , W 1 ). The same argument can be used for each k: as the (S k ) k split R + in disjoints intervals, then Q on each of these intervals is independent of Q of another interval.

In particular, U 1 1 = U 1 1 -S 0 , U 2 1 -S 1 (and all the following) are independent and can be dened as the rst jump of a Poisson point process on (0, +∞) × [0, ∞]. Then they follows an exponential law of parameter λ. Using time translation, we see that τ 1 , τ 2 (and so on) are dened the same way and follow the same law. Then W 1 , W 2 (and so on) are dened the same way and follow the same law. This construction indicates the renewal structure generated by the Hawkes process. We shall use this structure to prove limit theorems.

To this end remark that

N h t = ∞ i=1 1 U i ≤t = ∞ i=1 W i j=1 1 U j i ≤t . (3.2.6)
Introduce the renewal process associated to the S i 's

M h t := ∞ i=1
1 S i ≤t .

(3.2.7)

Since S i = i k=1 τ k we may introduce

N h t := ∞ i=1 W i 1 S i ≤t = M h t i=1 W i . (3.2.8)
For any t ∈ R + , the current window is the M h t + 1-th. N h t includes only the jumps up to the M h t -th window, while N h t can have more jumps. In particular,

N h t ≤ N h t ≤ N h t + W M h t +1 a.s.
(3.2.9)

We thus have

N h t = M h t i=1 W i + R h t 3.2.
Notation, denitions and results.

113

for some renewal process M h t and a remaining term R h t ≤ W M h t +1 , the W i 's being i.i.d.. Such processes are known as cumulative processes in the literature. A LLN and a CLT for N h t can be found in [1] theorems 3.1 and 3.2. The LD principle for cumulative processes is studied in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] in the special case

W i = F (τ i )
for some non-negative, bounded and continuous function F (see the references in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] for some previous results in still more specic cases). These results do not apply for Hawkes processes, and we had to establish a more general LD principle in the companion paper [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF]. In order to get similar results for N h t /t it will remain to study the remaining R h t .

LLN, CLT and asymptotic deviations for signed reproduction function h.

We now state the main results of the paper. The key is to get enough moments for the (τ i , W i ). i) For α < α 0 := min λ , ||h + || 1 -ln(||h + || 1 )-1

L(h)
we have E(e ατ 1 ) < +∞. ii) There exists θ 0 > 0 such that for θ < θ 0 , E(e θW 1 ) < +∞.

In particular τ 1 and W 1 have polynomial moments of any order.

The proof of this proposition is given in Section 3.4.1. Actually, one can give a lower bound on θ 0 . This lower bound diers whether h ≤ 0 or not. In the general case, the upper bound for θ 0 depends on a random variable S with distribution

P(S = k) = e -k h + 1 (k h + 1 ) k-1 k! .
Using a comparison with a queue process that will be detailed in the proof of Proposition 3.2.6 below one prove that θ 0 can be chosen as

θ 0 < ||h + || 1 -ln(||h + || 1 ) -1 and λ(E(e 2θ 0 S ) -1) < α 0 .
In the case of pure inhibition, i.e. h ≤ 0, the quantity ||h + || 1 -ln(||h + || 1 ) becomes innite. However using a comparison with a Poisson Process one can get another explicit bound for θ 0 , whose proof will also be given in Section 3.4.1.

Proposition 3.2.7. If h ≤ 0, one can choose θ 0 < -ln 1 -e -λL(h) in Proposition 3.2.6.

Remark 3.2.8. Exacts computations for moments of τ and W are dicult. Let us consider here and in section 3.3 some specic cases.

Chapter 3. Limit theorems for Hawkes processes

Notice that for h = 0 (i.e. in the case of a Poisson process), W 1 = 1 has exponential moments of any order and τ 1 whose distribution is exponential with parameter λ, has exponential moments up to order λ.

Another basic case is the canceling of intensity case, i.e. choosing the reproduction function as g = -λ 1 [0,A] for some positive λ and A. We have seen in Proposition 3.2.4 that the corresponding N g t is smaller than any N h t with L(h) = A.

Since for t ∈ (U 1 1 , U 1 1 +A) it holds Λ h (t) = 0, it immediately follows that τ 1 = U 1 1 +A and W 1 = 1, so that (W 1 , τ 1 ) ∼ (1, A + E(λ)) , so that E(τ 1 ) = A + λ -1 , Var(τ 1 ) = λ -2 , α 0 = λ and θ 0 = +∞. ♦
From these moments properties and the renewal structure of the Hawkes process, we will derive the following asymptotic results: Theorem 3.2.9 (Law of Large Numbers). Let h be a signed function satisfying Assumptions 3.2.3 and consider the Hawkes process N h given by (3.2.2). Then we have the following:

N h t t a.s. -→ t→∞ E[W 1 ] E(τ 1 )
.

Thanks to our comparison results and to (3.1.2) we have

λ 1 + λL(h) ≤ E[W 1 ] E(τ 1 ) ≤ λ 1 -||h + || 1 .
Our method will also provide us with a CLT.

√ t N h t t - E[W 1 ] E(τ 1 ) =⇒ t→∞ N (0, σ 2 ) with σ 2 = Var W 1 -τ 1 E[W 1 ] E(τ 1 ) E(τ 1 )
.

We nally state deviation results based on the results in the companion paper [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF], in which we obtain large deviation for general cumulative processes. To this end we need to introduce some notations. We also refer to [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] for a more general introduction on the topic. (

) 2
If θ 0 < +∞, we have for all a > 0 and κ ∈ (0, 1) lim sup x + (m -β)y + β ln 1 -x λ -βAx .

t→∞ 1 t ln P N h t t ≥ m + a ≤ -min inf z-m≥κa J(z) , 1 -κ 2 θ 0 a , ( 3
Notice that for a given x, sup y x + (m -β)y + β ln 1 -x λ -βAx < +∞ if and only if β = m due to the linear term in y. We deduce

J(m) = m Λ * 1 m , 1 .
It easily follows

J(m) = λ(1 -mA) -m + m ln m λ(1 -mA) .
♦ Remark 3.2.15. Added to the proof.

After completing the present paper and [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF], another proof of a LDP for cumulative processes was proposed in the preprint [START_REF] Zamparo | Large Deviation Principles for Renewal-Reward Processes[END_REF]. Some of the results in [START_REF] Zamparo | Large Deviation Principles for Renewal-Reward Processes[END_REF] should possibly complete the picture when θ 0 < +∞. ♦

One more example with explicit calculations: canceling intensity with delay.

We already discussed in Remark 3.2.8 the canceling of intensity case h = -λ 1 [0,A] .

In our second example we add a delay to the previous case: the inhibition only occurs after a lag period of length r > 0. Let λ > 0, r > 0 and A > r we consider h = -λ1 [r,r+A] . Then L(h) = r + A. We can again explicitly compute the law of W i and τ i . Let us rst remark that as r → 0 we recover the result of the canceling intensity case given in Remark 3.2.8.

Secondly we wonder whether one of both examples admits more points asymptotically. Therefore we are lead to study the ratio λ λA+1 λ(1+λr) λA+2λr+e -λr = λA + 2λr + e -λr (λA + 1)(1 + λr) ,

or equivalently the sign of

e -λr -1 + λr -λ 2 Ar = λ 2 r r 2 -A + ∞ k=3 (-λr) k k! .
using the series expansion of the exponential. We therefore deduce that since A > r, the right hand side is negative, and thus the ratio is less that 1.

Consequently, this proves that the lag induces asymptotically more points in the inhibited Hawkes process. Note that even if our proof only holds in the case of A > r since in this case there is a single time interval with canceled intensity between successive windows (see the proof ), we expect that delay in inhibition will also induce more points in more general framework.

Chapter 3. Limit theorems for Hawkes processes

Notice nally that ||h|| 1 is the same in the delayed and the non delayed case, therefore we deduce that the asymptotic proportion of points is not fully characterized by ||h|| 1 and that formula (3.1.2) cannot be extended to the inhibited setting.

♦

Proof of Proposition 3.3.1. Let us study Λ h on the time interval

[U 1 1 , U 1 1 + r + A]: for t ∈ [U 1 1 , U 1 1 + r)
, then for any u ∈ (0, t), t -u belongs to (0, t) and thus

Λ h (t) = λ + t 0 -λ1 (t-u)∈[r,r+A] N h (du) + = λ, for t ∈ [U 1 1 + r, U 1 1 + r + A], then Λ h (t) = λ + (0,t) -λ1 (t-u)∈[r,r+A] N h (du) + ≤ λ -λ1 (t-U 1 1 )∈[r,r+A] + = 0 .
From this, we deduce that all the points of N

h in ]U 1 1 , U 1 1 + r + A] actually belong to the interval ]U 1 1 , U 1 1 + r[. In particular, if N h has no points in ]U 1 1 , U 1 1 + r + A], then W 1 = 1 and τ 1 = U 1 1 + r + A. Let us now remark that N h (]U 1 1 , U 1 1 + r]
) follows a Poisson law of parameter λr since the intensity of Hawkes process is constant on this interval. In particular

N h ([U 1 1 , U 1 1 +r]) is nite almost surely. More generally for any 1 < k ≤ N h ([U 1 1 , U 1 1 + r]), then U 1 k ≤ U 1 1 + r and ∀t ∈ [U 1 k + r, U 1 k + r + A], Λ h (t) = 0.
Finally, since A > r we have that U 1 k + r ≤ U 1 1 + r + r ≤ U 1 1 + r + A, and thus the intensity remains null on the interval

[U 1 1 + r, U 1 k + r + A].
We can conclude that

W 1 = N h ([U 1 1 , U 1 1 + r]), (3.3.1)
τ 1 = U 1 W 1 + r + A. (3.3.2) Since the rst point in N h in the interval [U 1 1 , U 1 1 + r] is in U 1 1 we actually have W 1 = 1 + N h ((U 1 1 , U 1 1 + r)).
It follows that W 1 -1 ∼ P(λr) and E(W 1 ) = 1 + λr , Var(W 1 ) = λr and θ 0 = +∞ .

We nally study the law of τ 1 . From Equation (3.3.2), we can write

τ 1 = U 1 W 1 + r + A = r + A + U 1 1 + (U 1 W 1 -U 1 1 ),

One more example with explicit calculations: canceling intensity with delay. 119

where U 1 1 ∼ E(λ) by lemma 3.2.5 and U 1 1 and (U 1 W 1 -U 1 1 ) are independent. It remains to study the law of (U 1

W 1 -U 1 1 ). Thanks to (3.3.1), 0 ≤ U 1 W 1 -U 1 1 ≤ r. Let t ∈ [0, r],
we have:

P 0 ≤ U 1 W 1 -U 1 1 ≤ t = ∞ k=1 P {W 1 = k} ∩ {U 1 W 1 -U 1 1 ≤ t} .
For k = 1:

P {W 1 = 1} ∩ {U 1 W 1 -U 1 1 ≤ t} = P (W 1 = 1) = e -λr . For k > 1 since the intensity of the Hawkes process remains constant equal to λ on [U 1 1 , U 1 W 1 ] we can write U 1 W 1 -U 1 1 (law) = W 1 -1 k=1 T k
where (T k ) k∈N is a sequence of i.i.d E(λ). We can consider (T k ) k∈N as the interarrival times of a Poisson process of parameter λ coupled with our Hawkes process, as in Proposition 3.2.2. Then, T 0 = U 1 1 , and (T k ) k≥W 1 are dened. Then:

P ({W 1 = k} ∩{U 1 W 1 -U 1 1 ≤ t} = P 0 ≤ k-1 i=1 T i ≤ t ∩ T k + k-1 i=1 T i > r = E 1 0≤ k-1 i=1 T i ≤t P T k + k-1 i=1 T i > r | (T 1 , ..., T k-1 ) = E 1 0≤ k-1 i=1 T i ≤t e -λ(r-k-1 i=1 T i) = (R + ) k-1 1 0≤ k-1 i=1 s i ≤t λ k-1 e -λ k-1 i=1 s i × e -λ(r-k-1 i=1 s i ) ds 2 ...ds k = e -λr λ k-1 I k-1 (t) = e -λr (λt) k-1 (k -1)! . with I k (t) := (R + ) k 1 0≤ k i=1 s i ≤t ds 1 ...ds k = t k k! . Thus P 0 ≤ U 1 W 1 -U 1 1 ≤ t = e -λr + k≥2 e -λr (λt) k-1 (k -1)! = e -λ(r-t) .
Hence the distribution of U 1 W 1 -U 1 1 is given by e -λr δ 0 + λ e -λ(r-t) 1 (0,r] (t)dt.

Chapter 3. Limit theorems for Hawkes processes

An easy computation gives E(U 1

W 1 -U 1 1 ) = r -1 λ (1 -e -λr ). Finally we obtain that E(τ 1 ) = r + A + E(U 1 1 ) + E(U 1 W 1 -U 1 1 ) = r + A + λ -1 + r -λ -1 (1 -e -λr ) = 2r + A + λ -1 e -λr
From Theorem 3.2.9 we obtain the following LLN

N h t t a.s. -→ t→∞ 1 + λr 2r + A + λ -1 e -λr = λ(1 + λr) λA + 1 + (e -λr -1) + 2λr .
3.4 Proofs.

3.4.1 Proofs of Proposition 3.2.6 and 3.2.7

We start by proving that the random variables τ and W admit exponential moments.

Proof of Proposition 3.2.6.

Let h be a signed measurable function and h + its positive part. We generate N h and N h + by coupling as in Proposition 3.2.2. Recall that h + 1 < 1. We denote by W i , τ i , S i , ... (respectively W + i , τ + i , S + i , ... ) the renewal quantities associated to N h (resp. N h +

). Be careful that the previous construction of W + i , τ + i , S + i is done by using intervals of length L(h) not L(h + ). Notice that since L(h) ≥ L(h + ), then the renewal structure is well dened for

N h + . Moreover, if h ≤ 0, L(h + ) = 0, one can replace h + by h + ε = h + + ε 1 [0,L (h)
] and then let ε go to 0 in order to compare with [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF].

Thanks to Proposition 3.2.2, we have N h ≤ N h + a.s. We also know that U

1 1 = U +,1 1 . Moreover, τ 1 ≤ τ + 1 a.s. because the jumps of N h are included in those of N h + . We also have W 1 = N h ([0, τ 1 ]) ≤ N h ([0, τ + 1 ]) ≤ N h + ([0, τ + 1 ]) = W + 1 a.s. So W 1 ≤ W + 1 a.s.

Study of N h + :

First, we focus on N h + . According to [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF], we can associate a M/G/∞ queue to N h + . To do this, we consider:

Λ h + (t) = λ + (-L(h),t] h + (t -u)N h + (du).
We can consider the Hawkes process as the sum of: the arrivals of ancestors V k at rate λ and 3.4. Proofs. 121 a continuous time Galton-Watson process beginning at each V k with the following characteristics: the number of descendants follows a Poisson distribution with mean h + 1 and the times of births have the density h + / h + 1 .

In fact, to each arrival of an ancestor V k , we can associate a time H k corresponding to the life time of the cluster of V k . V k is independent of H k and the (H k ) k 's are independent.

We can associate to this process a queue in the following way:

the customers are the ancestors and arrive at rate λ, the service time for each customer is H k + L(h). We denote by Y t the number of customers in the queue at time t:

Y t = k 1 V k ≤t<V k +H k +L(h) . Let T + 1 = inf {t ≥ 0, Y t-= 0, Y t = 0}
, be the rst time the queue is empty. By proposition 2.6 of [START_REF] Costa | Renewal in Hawkes Processes with Self-Excitation and Inhibition[END_REF], we have:

∀α < α 0 := min λ, h + 1 -log( h + 1 ) -1 L(h) , it holds E[e αT + 1 ] < ∞ .
Of course λ > 0 and h + 1 -log( h + 1 ) -1 > 0, and so T + 1 admits an exponential moment.

Since τ + 1 is the rst time after U +,1 1 such that there were no jump during a time L(h). Thus τ + 1 = T + 1 and since τ 1 ≤ τ + 1 , part (i) of the proposition is proved.

In order to prove (ii) it is enough to show that the distribution of W + 1 admits exponential moments. Recall that

W + 1 = N h + ([0, τ + 1 ]) .
According to [START_REF] Bordenave | Large Deviations of Poisson Cluster Processes[END_REF] (see proof of Theorem 3.2 and proof of Theorem 3.4 therein),

lim t→+∞ 1 t ln E e θN h + ([0,t]) = λ(E(e θS ) -1) := µ(θ) < +∞ as soon as θ < h + 1 -log( h + 1 ) -1.
Here S is distributed according to (see [START_REF] Bacry | Hawkes Model for Price and Trades High-Frequency Dynamics[END_REF] in [START_REF] Bordenave | Large Deviations of Poisson Cluster Processes[END_REF])

P(S = k) = e -k h + 1 (k h + 1 ) k-1 k! .
It is thus immediate that µ(θ) goes to 0 as θ goes to 0.

For ε > 0 we may thus choose θ small enough such that α 0 -2ε ≥ µ(2θ) + ε .

For this θ, one can nd t θ such that for t ≥ t θ , E e 2θN h + ([0,t]) ≤ e t(µ(2θ)+ε) .

Chapter 3. Limit theorems for Hawkes processes

It follows

E(e θW + 1 ) = E e θN h + ([0,τ + 1 ]) ≤ ∞ k=1 E e θN h + ([0,k]) 1 k-1≤τ + 1 <k ≤ ∞ k=1 β k E e 2θN h + ([0,k]) + 1 β k P(k -1 ≤ τ + 1 ) ≤ A(t θ ) + ∞ k=[t θ ]+1 β k e k(µ(2θ)+ε) + E(e (α 0 -ε)τ + 1 ) β k e -(k-1)(α 0 -ε)
where A(t θ ) denotes the nite sum up to k

= [t θ ]. Choosing β k = k -2 e -k(µ(2θ)+ε)
the k'th term of the remaining sum is smaller than 1/k 2 +c k 2 e -ε(k-1) and the series is thus convergent. Since ε is arbitrary, (ii) follows.

Proof of Proposition 3.2.7.

We consider a process N h , generated by the Poisson point process Q, as in the Proposition 3.2.2. Since h ≤ 0, we will couple (and upper-bound) this time the Hawkes process with the Poisson point process R on R + , with intensity λ, generated by the same Poisson point process Q on (0, ∞) 2 . Since ∀t ≥ 0 λ ≥ Λ h (t) a.s.

we deduce that R ≥ N h .

We can now upper bound the length of the rst time window τ 1 by a similar quantity associated with R. Recall that U 1 1 is the rst jump time of N h and dene:

τ = inf{t > U 1 1 , R[t -L(h), t) = 0, R(t -L(h), t] = 0}. (3.4.1)
τ indicates the rst moment such that there were no jump for the process R during an interval of length L(h). In particular, there weren't jump for N h either. Therefore τ 1 ≤ τ a.s. and

W 1 = N h ([U 1 1 , τ 1 ]) ≤ R([U 1 1 , τ 1 ]) ≤ R([0, τ ]).
It is thus enough to get an upper bound for E e θ R([0,τ ]) .

To this end we shall study the random variable τ . Denote by V i the jumps of the Poisson point process R. From the denition there exists a random integer K such that

τ = V K + L(h). The denition of K leads to K = R[0, τ ].

Proofs. 123

From the independence of the times between jumps of R we deduce that

P(K = 1) = P[τ = V 1 + L(h)] = P[V 2 -V 1 ≥ L(h)] = e -λL(h) , P(K = 2) = P[τ = V 2 + L(h)] = P[{V 2 -V 1 < L(h)} ∪ {V 3 -V 2 ≥ L(h)}] = P[V 2 -V 1 < L(h)]P[V 3 -V 2 ≥ L(h)] = (1 -e -λL(h) ) e -λL(h) , ∀k ≥ 2, P(K = k) = P[τ = V k + L(h)] = (1 -e -λL(h) ) k-1 e -λL(h) .
K is a geometric random variable with parameter e -λL(h) and thus admits exponential moments provided e θ (1 -e -λL(h) ) < 1 which concludes the proof.

Proof of the LLN and CLT

Proof of Theorem 3.2.9 and Theorem 3.2.10.

Recall that

N h t ≤ N h t ≤ N h t + W M h t +1 a.s.
where

N h t := ∞ i=1 W i 1 S i ≤t = M h t i=1 W i and M h t := ∞ i=1 1 S i ≤t ,
as explained in (3.2.6), (3.2.7), (3.2.9).

As we previously said Theorem 3.1 and Theorem 3.2 Chapter 6 in [START_REF] Asmussen | Applied Probability and Queues. 2nd ed. Stochastic Modelling and Applied Probability[END_REF] furnish

N h t t a.s. -→ t→∞ E[W 1 ] E(τ 1 ) and √ t N h t t - E[W 1 ] E(τ 1 ) =⇒ t→∞ N (0, σ 2 ) with σ 2 = Var W 1 -τ 1 E[W 1 ] E(τ 1 ) E(τ 1 )
.

It is thus enough to control the remaining (or error) term W M h t +1 i.e to prove lim t→∞ Actually we will prove stronger results. Let β(n) an increasing sequence going to innity and ε > 0. Introduce the independent events

W M h t +1 t = 0 a.s., (3.4.2
A n = {W n > εβ(n)}. Then lim sup n A n = {lim sup n Wn β(n) > ε}. Since the (W i ) i are nite i.i.d random variables n P(A n ) = n P (W n > εβ(n)) = n P(W 1 > εβ(n)) .
Thanks to Proposition 3.2.6 and to Markov inequality, we know that for

P(W 1 > εβ(n)) ≤ E[e θ 0 W 1 ] e -θ 0 ε β(n) .
We may now apply Borel-Cantelli, telling that provided n e -θ 0 ε β(n) < +∞,

P(lim sup n A n ) = 0.
The previous holds with β(n) = n α for any α > 0. We have proved in particular that W n √ n a.s.

-→ n→∞ 0 .

Since M h t is a non-decreasing family of integers going to innity almost surely,

W M h t +1 M h t + 1 a.s.
-→ t→∞ 0 .

It remains to recall that

M h t + 1 t -→ t→∞ 1 E(τ 1 ) a.s., (3.4.4)
to conclude that

W M h t +1 √ t a.s.
-→ t→∞ 0 .

Proof of Corollary 3.2.13

Proof of Corollary 3.2.13.

In order to prove the rst part of Corollary 3.2.13 it is enough to show that N h t /t and N h t /t are exponentially equivalent, i.e. that for each δ > 0, lim sup

t→∞ 1 t ln P N h t t - N h t t > δ = -∞ .

Proofs. 125

To this end it is enough to show that for all δ > 0 lim sup

t→∞ 1 t ln P W M h t +1 > δt = -∞ . (3.4.5)
We will decompose the probability space into two events: M h t ≤ t 2 and M h t > t 2 .

It holds

P W M h t +1 > δ t ≤ P M h t > t 2 + P W M h t +1 > δt ∩ M h t ≤ t 2 ≤ P M h t > t 2 + P ∃k ∈ {1, ..., t 2 + 1 }, W k > δt ∩ M h t ≤ t 2 + 1 ≤ P M h t > t 2 + P ∃k ∈ {1, ..., t 2 + 1 }, W k > δt ≤ P M h t > t 2 + t 2 +1 j=1 P (W k > δt) ≤ P M h t > t 2 + (t 2 + 1)P (W 1 > δt) .
On one hand, we have, by Markov's inequality, for all θ 0 > θ > 0,

P (W 1 > δt) ≤ E[e θW 1 ] e -θδt so that lim sup t→+∞ 1 t ln(1 + t 2 ) P (W 1 > δt) ≤ -θ 0 δ.
On the other hand, according to [START_REF] Tiefeng | Large Deviations for Renewal Processes[END_REF] Theorem 2.3, for all x > 0: lim sup ≤ -θ 0 δ .

t→+∞ 1 t ln P M h t t ≥ x ≤ -J τ 1 (x), where J τ 1 (x) = sup η {η -x ln E[e ητ 1 ]}. Since η → E(e ητ 1 ) is continuous on R -there exists some η 0 such that E(e η 0 τ 1 ) = e -1 . It follows J τ 1 (x) ≥ η 0 + x. Choose t 1 ,
This completes the proof for θ 0 = +∞.

Let us now assume θ 0 < ∞. Recall that m = E(W 1 ) E(τ 1 ) , then (3.2.10) is a conse- quence of the following line of reasoning:

P N h t t ≥ m + a ≤ P N h t t + W M h t +1 t ≥ m + a ≤ P N h t t ≥ m + κ 1 a + P W M h t +1 t ≥ (1 -κ 1 )a
where κ 1 ∈ (0, 1), yielding lim sup

t 1 t ln P N h t t ≥ m + a ≤ max lim sup t 1 t ln P N h t t ≥ m + κ 1 a , lim sup t 1 t ln P W M h t +1 t ≥ (1 -κ 1 )

a

Now applying Theorem 3.2.12 with κ 2 and (1 -κ 2 ), we deduce that lim sup As we recalled in the introduction, in the linear case the LLN, the CLT and the LDP are completely characterized by ||h|| 1 . As we have shown in Section 3.3, the almost linear case with inhibition we are looking at is dramatically dierent, since the limiting behaviour is not fully determined by ||h|| 1 nor even by some moments of h. The renewal description of the Hawkes process we have used allows us to characterize all these limit theorems in terms of the joint law of (τ 1 , W 1 ). It should be very interesting to link this distribution with h. As for the non linear self-excited case such a goal seems dicult to reach.

t 1 t ln P N h t t ≥ m + a ≤ max - inf z-m≥κ 2 κ 1 a J(z) , -(1 -κ 2 )κ 1 θ 0 a , - (1 -
Another interesting direction should be to obtain non asymptotic deviation bounds (or concentration properties). Since the Large Deviation Principle for cumulative processes we have proved in [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF] is based on the contraction of a higher level LDP, new methods are necessary for non asymptotic results.

The methods of the paper can be used for more general jump rate functions f , provided one can generalize the construction of the sequence (τ i , W i ). This generalization is partly done in [START_REF] Graham | Regenerative Properties of the Linear Hawkes Process with Unbounded Memory[END_REF] in which a regenerative structure is exhibited without the assumption of bounded support for the reproduction function h and in [START_REF] Bonde | Renewal Time Points for Hawkes Processes[END_REF] which exhibit renewal points for non linear Hawkes processes and age-dependent Hawkes processes. Understanding the brain activity is both a complex and important challenge in current research. Of course, interests are plentiful: characterizing brain functions, understanding structures and links between them and guring out some phenomena -such as cyclic heartbeat. A way of modeling this activity is by considering a very large number of individual neurons with interactions. Since the number of neurons in a human brain is around 10 11 , and even "small" parts of the brain are thus constituted of very large number of them, such a strategy can be considered coherent.

The main quantity we study is the membrane potential of the nerve cells: it can "easily" be observed and its modication characterizes a synapse (an interaction between neurons). Neurons regulate their electrical potential. In general, without interaction, the potential evolves with time but has quite small changes. Incoming potentials from other neurons are usually what make the neuron re, i.e. send potential to other neurons. We will here focus on an homogeneous network of neurons and consider mean-eld interactions. This way, each neuron will interact with every other one, as it can be the case in small regions of the brain. The parameters of the model will be considered the same for each neuron.

A classical model was introduced by Hodgkin and Huxley [START_REF] Hodgkin | A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve[END_REF] The deterministic FitzHugh-Nagumo model for one neuron (or one particle) is given by the following equations:

dX t = (X t -(X t ) 3 -C t -α)dt dC t = (γX t -C t + β)dt,
where X is the potential membrane and C is a recovery variable, called the adaptation variable. The parameters γ and β are positive constants that determine the duration of an excitation and the position of the equilibrium point of this system.

Finally α ∈ R is the magnitude of a stimulus current (an entrance current in the system). Note that the variable C isn't a physical quantity, and is used to allow X to mimic the behavior of the potential. This variable C has linear dynamics and provides a slower negative feedback.

Stochastic versions of the model, adding some noise on one or both coordinates have also been introduced. Noise models dierent types of randomness depending on the coordinates: when the noise is in the rst equation (dynamics of X) with a standard deviation σ X > 0, it models a noisy presynaptic current. When it is in the second equation (dynamics of C) with a standard deviation σ C > 0, it describes a noisy conductance dynamic (a noise in the chemical behavior). In general, noise in this model is additive. Various mathematical questions can be studied. Some authors choose to focus on the properties of the natural macroscopic limit of the model as N → ∞ when it is clearly dened (see system (4.1.2)), when others work on properties of the particles system for xed N . These models can be quite complicated to study mathematically. The main objectives are to characterize the behavior of these models when the number of neurons N tends to +∞ in a mean-eld limit, and to prove whether or not there exists an equilibrium, a stationary behavior, when t tends to +∞. The question of the synchronization of neurons can also be studied, since it is a phenomenon observed in dierent contexts, such as the generation of respiratory rhythm or complex neurological functionalities.

Some authors study a model with two noises: in this case, the stochastic system is non-degenerate and is then elliptic.

Berglund and Landon describe the behavior of the deterministic FitzHugh-Nagumo model for one neuron in [START_REF] Berglund | Mixed-Mode Oscillations and Interspike Interval Statistics in the Stochastic FitzHugh-Nagumo Model[END_REF], and consider the stochastic model to work on the behavior of the interspike interval and the distribution of oscillations of the solution. In [START_REF] Bemmo | Combined Eects of Correlated Bounded Noises and Weak Periodic Signal Input in the Modied FitzHughNagumo Neural Model[END_REF], Tatchim Bemmo, Siewe Siewe and Tchawoua focus on a quite dierent stochastic model by considering additive noise η on the dynamics of X, and multiplicative noise ξ on the dynamics of C, both dened as sinusoidal function of correlated Brownian motions. They choose to avoid Gaussian noises since it is an unbounded noise.

They also consider a deterministic and periodic entrance signal in the rst equation.

They observe abrupt transitions of the potential membrane X when the intensity of the noise is gradually changed.

In general, a lot of authors focus on a noise on only one variable, thus the stochastic model is degenerate. In [START_REF] Henry | Determination of Firing Times for the Stochastic Fitzhugh-Nagumo Neuronal Model[END_REF], the authors work on the determination of ring time. They consider a stochastic FitzHugh-Nagumo model for one neuron, with Brownian noise on X, obtain approximation of ring times and compare them with numerical simulations. Thieullen describes the behavior of the solution of one deterministic FitzHugh-Nagumo system and extends the result in the case of a stochastic FitzHugh-Nagumo system, in [START_REF] Thieullen | Deterministic and Stochastic FitzHughNagumo Systems[END_REF]. In her case, the stochasticity is a Brownian noise on the dynamics of X. In [START_REF] José | Hypoelliptic Stochastic Fitzhugh-Nagumo Neuronal Model: Mixing, up-Crossing and Estimation of the Spike Rate[END_REF], León and Samson consider a FitzHugh-Nagumo model with a noise on C but not on X, i.e. σ X = 0, and study the properties of the equations for one neuron. In particular, they focus on hypoellipticity of the model, the existence and uniqueness of an invariant probability model. A second spatial derivative of X is added in the dynamics of X. [START_REF] Lv | Limit Dynamics for the Stochastic FitzHugh-Nagumo System[END_REF], [START_REF] Li | Dynamics for Stochastic Fitzhugh-Nagumo Systems with General Multiplicative Noise on Thin Domains[END_REF], [START_REF] Li | Limiting Dynamics for Stochastic Fitzhugh-Nagumo Equations on Large Domains[END_REF] and [START_REF] Li | Regular Dynamics for Stochastic FitzHugh-Nagumo Systems with Additive Noise on Thin Domains[END_REF] explore the notion of random attractors in these models. Numerical schemes for the interacting particles system are also studied, for instance in [START_REF] Reisinger | An Adaptive Euler-Maruyama Scheme for Mckean-Vlasov SDEs with Super-Linear Growth and Application to the Mean-Field Fitzhugh-Nagumo Model[END_REF].

Framework and results

Combining noise and interaction, we work specically on the following equations, for 1 ≤ i ≤ N , where N is the number of neurons:

dX i,N t = (X i,N t -(X i,N t ) 3 -C i,N t -α)dt + 1 N N j=1 K X (Z i,N t -Z j,N t ) + σ X dB i,X t dC i,N t = (γX i,N t -C i,N t + β)dt + 1 N N j=1 K C (Z i,N t -Z j,N t ) + σ C dB i,C t , (4.1.1)
where we denote by Z i t the couple (X i t , C i t ) to simplify the notations. We assume (B i,X t ) i and (B i,C t ) to be independent Brownian motions. Here, we consider two Brownian noises B X and B C , one on each equation, and thus assume that each neuron has its own independent noises, and that there is no environmental (or shared) noise.

We also assume K X and K C to be Lipschitz continuous and respectively denote their Lipschitz constants by L X and L C . The goal of this article is to describe the behavior of this network as the number N of neurons tends to innity.

To describe its behavior, we consider the R 2 -valued process ( Zt ) t≥0 = ( Xt , Ct ) t≥0 evolving according to the following non-linear stochastic dierential equation of

McKean-Vlasov

type d Xt = ( Xt -( Xt ) 3 -Ct -α)dt + K X * μt ( Zt )dt + σ x d BX t d Ct = (γ Xt -Ct + β)dt + K C * μt ( Zt )dt + σ c d BC t , (4.1.2)
where μt = Law( Zt ) is the law at time t of the process ( Xt , Ct ), and * denotes the operation of convolution, i.e.

K * ν(u) = K(u -v)ν(dv).
To some extent, (4.1.1) can be seen as an approximation of (4.1.2) in which the operation of convolution is applied to the empirical measure µ t,emp = 1

N N i=1 δ Z i t .
We wish to prove that the law µ N t of the network (4.1.1) converges in some sense to μ⊗N t (i.e the law of the solution of (4.1.2) tensorized N times) as N tends to innity. This phenomenon has been stated under the name propagation of chaos -an idea motivated by M. Kac [START_REF] Kac | Foundations of Kinetic Theory[END_REF]) and greatly developed by A.S. Sznitman [START_REF] Sznitman | Topics in Propagation of Chaos[END_REF]-as it amounts to saying that, as the number of particles increases in the system, two particles will become "more and more" independent, converging towards a tensorized law. The notion of "propagation" refers to the fact that proving such convergence 134 Chapter 4. Propagation of chaos for FHN neurons at time 0 is sucient to prove it at a later time t.

In order to prove the convergence of µ N t to μ⊗N t , we follow the coupling method described in a recent work by one of the authors in [START_REF] Guillin | Convergence Rates for the Vlasov-Fokker-Planck Equation and Uniform in Time Propagation of Chaos in Non Convex Cases[END_REF], the result of which cannot be applied directly here. This method has been put forward by Eberle, following earlier works by Lindvall and Rogers [START_REF] Lindvall | Coupling of Multidimensional Diffusions by Reection[END_REF]. Let us briey recall the method.

We consider

r i t = | Xi t -X i,N t | + δ| Ci t -C i,N t | with δ > 0, a constant not yet specied.
A natural distance between probability measures is the Wasserstein distance, linked to the theory of optimal transport. For µ and ν two probability measures on R d , we denote

W p (µ, ν) = inf X∼µ, Y ∼ν E ||X -Y || p p 1/p , (4.1.3)
where || • || p denotes the usual L p distance on R d . It is thus dened as the minimum over all possible choices of a pair (X, Y ), such that X is distributed according to µ and Y according to ν, of the expectation of the distance between X and Y . The basic idea behind a coupling method is then that an upper bound on the Wasserstein distance between µ and ν is given by the construction of any pair of random variables distributed according to these probability measures. Thus, instead of considering the minimum over all possible coupling, we construct simultaneously two solutions of (

2) that will tend to get closer together as the number of neurons increases.

Have

Xi t , Ci t , for i between 1 and N , be N independent copies of a solution of (4.1.2) driven by some independent Brownian motions ( Bi,X t ) t 0 and ( Bi,C t ) t 0 . A coupling of Xi t , Ci t and X i,N t , C i,N t then follows from a coupling of the Brownian motions B and B.

The rst natural choice, popularized by Sznitman [START_REF] Sznitman | Topics in Propagation of Chaos[END_REF], is the synchronous coupling and consists in choosing B = B. By doing so, when considering the time evolution of Zi

t -Z i,N t = Xi t -X i,N t , Ci t -C i,N t
, the noise cancels out. The contraction of a distance between the processes can then only be induced by the deterministic drift, as in [START_REF] Bolley | Trend to Equilibrium and Particle Approximation for a Weakly Selfconsistent Vlasov-Fokker-Planck Equation[END_REF], and this usually only holds under rather restrictive conditions (in particular the drift should be strongly convex). Nevertheless, in our case, the calculation of the evolution of Xi t -X i,N t and Ci t -C i,N t (see later) shows that there is still some deterministic contraction when Xi t -X i,N t = 0. We can therefore use a synchronous coupling in the vicinity of this subspace.

Outside of this subspace, we use the noise to get the processes closer together.

In the direction orthogonal to this space we consider B = -B, as this maximizes the variance of the noise. This yields the use of the reection coupling of [START_REF] Lindvall | Coupling of Multidimensional Diffusions by Reection[END_REF]. Notice however at this stage that, because of the symmetry of the noise, there is a priori no reason r i t should decrease rather than increase. This invites us to consider f (r i t ), with f a concave function, so that a random decrease has more eect than a random increase of the same value. We will dene the function f later.

Finally we construct a Lyapunov function H to take into account the trend of each process to come back to some compact set of R 2 . We are then led to the study of a suitable distance between the two processes, which will be of the form

ρ t := 1 N N i=1 f (r i t )(1+ H( Zi t )+ H(Z i,N t ))
, where > 0. This quantity controls the usual L 1 and L 2 distances between the two systems, and is interesting as, when r i t is small, f (r i t ) tends to decrease either because of the deterministic drift or the reection coupling, and when r i t is big, the Lyapunov functions H will tend to decrease. We thus show that Eρ t decays exponentially fast. This leads to several constraints on δ, and on the parameters involved in the denition of f , and we have to prove that it is possible to meet all these conditions simultaneously. In reality, the quantity ρ t considered will be a slight twist of the one given above (see (4.2.26)) so as to take into account the non linearity of the process.

As explained, this method requires some noise in the direction orthogonal to the naturally contracting subspace. This means, in the description of the method above, that one should have σ X > 0 (so that we can use a reection coupling to bring Xi t and X i,N t closer together). In the case σ X = 0 and σ C > 0, a modication of the calculations is necessary. We describe this case in a separate section. Assumption 4.1.1. Let L X,max and L C,max be two universal constants

∃L X ∈ [0, L X,max ], ∀z, z ∈ R 2 |K X (z) -K X (z )| ≤ L X ( z -z 1 ) ∃L C ∈ [0, L C,max ], ∀z, z ∈ R 2 |K C (z) -K C (z )| ≤ L C ( z -z 1 )
K X (0, 0) = 0 and K C (0, 0) = 0 These constants L X,max and L C,max might seem at rst glance o putting as they are not given. When we prove non uniform in time propagation of chaos, these constants can be chosen to be +∞. When we prove uniform in time propagation of chaos however, L X,max and L C,max are a priori bounds : Theorem 4.1.4 below will be true for L X and L C suciently small. The condition L X ≤ L X,max and L C ≤ L C,max are therefore not restrictive conditions, and are useful in proving some parameters are independent of L X and L C . Remark 4.2.7 below shows that one can for instance consider L X,max = 4 and L C,max = 1 5 .

Before any result on propagation of chaos, we prove that both systems (4. There exists a unique and strong solution for system (4.1.1) and a unique and strong solution for system (4.1.2). Now, we can state our results on propagation of chaos. We denote W 1 and W 2 the usual L 1 and L 2 Wasserstein distances.

W 1 µ k,N t , μ⊗k t ≤ C 1 e C 2 t k √ N ,
for all k ∈ N, where µ k,N t is the marginal distribution at time t of the rst k neurons (X 1 t , C 1 t ), ...., (X k t , C k t ) of an N particles system (4.1.1) with initial distribution (µ 0 ) ⊗N , while μt is a solution of (4.1.2) with initial distribution µ 0 .

Our main result consist in removing the time dependency in the previous upperbound. This uniform propagation of chaos requires stronger assumptions on the interaction kernels. Theorem 4.1.4. [Uniform in time propagation of chaos] Let C0 > 0 and ã > 0. There is an explicit C K X ,K C > 0 such that, for all K X and K C satisfying Assumptions 4.1.1 with L X , L C < C K X ,K C , there exist explicit B 1 , B 2 > 0, such that for all probability measures µ 0 on R 2 satisfying E µ 0 e ã(|X|+|C|) ≤ C0 ,

W 1 µ k,N t , μ⊗k t ≤ B 1 k √ N , W 2 2 µ k,N t , μ⊗k t ≤ B 2 k √ N ,
for all k ∈ N, where µ k,N t is the marginal distribution at time t of the rst k neurons (X 1 t , C 1 t ), ...., (X k t , C k t ) of an N particles system (4.1.1) with initial distribution (µ 0 ) ⊗N , while μt is a solution of (4.1.2) with initial distribution µ 0 .

C K X ,K C is explicitly given, depending on other parameters in 4.2.24. It's a non optimal denition. As previously mentioned, this work follows the method described in [START_REF] Guillin | Convergence Rates for the Vlasov-Fokker-Planck Equation and Uniform in Time Propagation of Chaos in Non Convex Cases[END_REF]. Beyond the result of uniform in time propagation of chaos for the FitzHugh-Nagumo model, which is in itself an interesting result, the present work is also a testament to the robustness of the coupling method.

Existence of solutions

First of all, we prove Proposition 4.1.2, i.e existence of strong solutions of systems Let's denote, for K ∈ R + ,

g K (x) =    -K 3 if x < -K x 3 if x ∈ [-K, K] K 3 if x > K. g K is locally Lipschitz and is bounded.
Thus, it's well known (see Chapter 3 [START_REF] Ikeda | Stochastic Dierential Equations and Diusion Processes[END_REF]) that the following system (under Assumption 4.1.1)

dX i,N t = (X i,N t -g K (X i,N t ) -C i,N t -α)dt + 1 N N j=1 K X (Z i,N t -Z j,N t ) + σ X dB i,X t dC i,N t = (γX i,N t -C i,N t + β)dt + 1 N N j=1 K C (Z i,N t -Z j,N t ) + σ C dB i,C t , (4.1.4)
for 1 ≤ i ≤ N , have strong and unique solution.

In consequence, for a xed K ∈ R + , there exists a strong solution of system (4.1.1) until time T K = sup{t, ∀i, ∀s ≤ t, X i t ≤ K and C i t ≤ K}, and the solution coincide with the solution of the system with g K .

We have the following Proposition:

Proposition 4.1.5. If, for each i ≤ N , E(|X i,N 0 | 2 ) < +∞ and E(|C i,N 0 | 2 ) < +∞, then there exists C t < ∞ such that, for each i ≤ N : E |X i,N t | 2 + |C i,N t | 2 ≤ C t .
T ∞ = inf{t, ∃i, ∀A > 0, ∃ε > 0, ∀s ∈ (t -ε, t), X i,N s > A or C i,N s > A}
we deduce ∀t ∈ R + , P(T ∞ ≤ t) = 0. Eventually, there exists unique and strong solution for system (4.1.1).

The existence and unicity of a solution of (4.1.2) is known from the Theorem 3.3 from [START_REF] Gonçalo Dos Reis | FreidlinWentzell LDP in Path Space for McKeanVlasov Equations and the Functional Iterated Logarithm Law[END_REF]. We only have to prove that the Assumptions 3.2 [START_REF] Gonçalo Dos Reis | FreidlinWentzell LDP in Path Space for McKeanVlasov Equations and the Functional Iterated Logarithm Law[END_REF] are veried. We dene, for all t ∈ R + , z = (x, c) ∈ R 2 and for all probability distribution ν with a nite variance:

b(t, z, ν) = x -x 3 -c -α + K X * ν(z) γx -c + β + K C * ν(z) σ(t, z, ν) = σ X σ C .
σ is a constant function, so it clearly satises the dierent conditions.

For t ∈ R + , z, z in R 2 , and ν a probability measure:

z -z ,b(t, z, ν) -b(t, z , ν) =(x -x ) (x -x ) -(x 3 -x 3 ) -(c -c ) + K X * ν(z) -K X * ν(z ) + (c -c ) γ(x -x ) -(c -c ) + K C * ν(z) -K C * ν(z ) =(x -x ) 2 -(x -x ) 2 (x 2 + xx + x 2 ) + (γ -1)(c -c )(x -x ) -(c -c ) 2 + (K X * ν(z) -K X * ν(z )) + (K C * ν(z) -K C * ν(z )).
Since (x 2 +xx +x 2 ) ≥ 0, the second term is non-positive. K X and K C are Lipschitz function, so the last line is clearly bounded by z -z 2 2 up to a multiplicative constant. Then, there exists a constant L such that

z -z , b(t, z, ν) -b(t, z , ν) ≤ L z -z 2 2 .
Since K X and K C are Lipschitz function, we also have, for all probability distribution ν and ν with a nite variance, b(t, z, ν) -b(t, z, ν ) 2 ≤ LW 2 (ν, ν ).

Eventually, since b is Locally Lipschitz with polynomial growth, each Assumption is satised and Theorem 3.3 [START_REF] Gonçalo Dos Reis | FreidlinWentzell LDP in Path Space for McKeanVlasov Equations and the Functional Iterated Logarithm Law[END_REF] can be applied. Note that we could also apply Proposition 2.19 from [START_REF] Luçon | Mean Field Limit for Disordered Diusions with Singular Interactions[END_REF] : Assumptions are the same, and it gives a result for interaction depending on a spatial position.

To complete the Proposition 4.1.5, we also give the following

E | Xt | 2 + | Ct | 2 ≤ C 0,1 e C 0,
dX i,N t = (X i,N t -(X i,N t ) 3 -C i,N t -α)dt + 1 N N j=1 K X (Z i t -Z j t )dt + σ x dB i,X t dC i,N t = (γX i,N t -C i,N t + β)dt + 1 N N j=1 K C (Z i t -Z j t )dt + σ c dB i,C t and d Xi t = ( Xi t -( Xi t ) 3 -Ci t -α)dt + K X * μt ( Zi t )dt + σ x dB i,X t d Ci t = (γ Xi t -Ci t + β)dt + K C * μt ( Zi t )dt + σ c dB i,C t ,
with μt the law of Z1

t . The method is the following :

we compute the time evolution of E(r i t ) using Ito's formula, we control the dierence between the drifts

1 N j =i K( Zi t -Zj t ) and K * μ t ( Zi t )
using some form of law of large number. This is where the convergence rate √ N appears, and we conclude using Gronwall's lemma.

Time evolution : We have,

d(X i,N t -Xi t ) = (X i,N t -Xi t ) -(X i,N t ) 3 -( Xi t ) 3 -(C i,N t -Ci t ) + 1 N N j=1 K X (Z i t -Z j t ) -K X * μt ( Zi t )   dt.
We denote sign(x) =

x |x| if x = 0, 0 otherwise, and obtain, using Ito's formula,

d|X i,N t -Xi t | = sign(X i,N t -Xi t )(X i,N t -Xi t ) -sign(X i,N t -Xi t ) (X i,N t ) 3 -( Xi t ) 3 -sign(X i,N t -Xi t )(C i,N t -Ci t ) +sign(X i,N t -Xi t ) 1 N N j=1 K X (Z i t -Z j t ) -K X * μt ( Zi t )   dt ≤ |X i,N t -Xi t | -(X i,N t ) 3 -( Xi t ) 3 + C i,N t -Ci t + 1 N N j=1 K X (Z i t -Z j t ) -K X * μt ( Zi t )   dt. (4.1.7)
Similarly,

d(C i,N t -Ci t ) = γ(X i,N t -Xi t ) -(C i,N t -Ci t ) + 1 N N j=1 K C (Z i t -Z j t ) -K C * μt ( Zi t )   dt,
and we obtain

d|C i,N t -Ci t | ≤ γ X i,N t -Xi t -|C i,N t -Ci t | + 1 N N j=1 K C (Z i t -Z j t ) -K C * μt ( Zi t )   dt. (4.1.8) Thus, denoting r i t = |X i,N t -Xi t | + |C i,N t -Ci t | dr i t ≤ (1 + γ)|X i,N t -Xi t | -(X i,N t ) 3 -( Xi t ) 3 + 1 N N j=1 K X (Z i t -Z j t ) -K X * μt ( Zi t ) + 1 N N j=1 K C (Z i t -Z j t ) -K C * μt ( Zi t )   dt.
Since the cubic term is negative, we obtain

dr i t ≤   (1 + γ)r i t + 1 N N j=1 K X (Z i t -Z j t ) -K X * μt ( Zi t ) + 1 N N j=1 K C (Z i t -Z j t ) -K C * μt ( Zi t )   dt.
Dierence of the drifts : Let us now consider these last two terms

1 N N j=1 K X (Z i t -Z j t ) -K X * μt ( Zi t ) ≤ 1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) + 1 N N j=1 K X (Z i t -Z j t ) - 1 N N j=1 K X ( Zi t -Zj t ) .
The rst sum can be decomposed, using Assumption 4.1.1,

1 N N j=1 K X (Z i t -Z j t ) -K X ( Zi t -Zj t ) ≤ L X N N j=1 Z i t -Z j t -( Zi t -Zj t ) 1 ≤ L X N N j=1 Z i t -Zi t 1 + Z j t -Zj t 1 ≤L X r i t + L X N N j=1 r j t .
Similarly, we obtain

1 N N j=1 K C (Z i t -Z j t ) -K C * μt ( Zi t ) ≤ L C r i t + L C N N j=1 r j t + 1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t ) .
Hence, we get

dr i t ≤   (1 + γ)r i t + (L X + L C )   r i t + 1 N N j=1 r j t   + 1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) + 1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t )   dt.
By considering the expectation, since E(r j t ) = E(r i t ) for each j, by exchangeability of the particles, we have

dE(r i t ) ≤ (1 + γ + 2L X + 2L C )E(r i t ) + E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t )   +E   1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t )     dt.
Now, we bound the interaction part. We begin with K X . By Cauchy-Schwarz, we can write

E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t )   ≤ E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2   1/2
We notice that ( Zj t ) j are i.i.d with law μt . Let's denote Zt a generic random variable of law μt independent of Zi t . What is more,

K X * μt ( Zi t ) = K X ( Zi t -z)μ t (dz) = E[K X ( Zi t -Zt )| Zi t ]
.

Hence E   E   1 N -1 j =i K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2 Zi t     = E   Var   1 N -1 j =i K X ( Zi t -Zj t ) Zi t     = 1 N -1 E Var K X ( Zi t -Zt ) Zi t ≤ L 2 X N -1 E Var Zi t -Zt 1 Zi t . Since E Var Zi t -Zt 1 Zi t ≤E E Zi t -Zt 2 1 Zi t ≤E E 2 Zi t 2 1 + 2 Zt 2 1 Zi t ≤ 4E( Zt 2
1 ),

we obtain

E   E   1 N -1 j =i K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2 Zi t     ≤ 4L 2 X N -1 E( Zt 142
Chapter 4. Propagation of chaos for FHN neurons

We now want to control E 1

N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2 . We decompose it with E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2   =E   N -1 N 1 N -1 N j=1 K X ( Zi t -Zj t ) - N -1 N + 1 N K X * μt ( Zi t ) 2   ≤2 N -1 N 2 E   1 N -1 N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2   + 2 N 2 E |K X * μt ( Zi t )| 2 . Since E |K X * μt ( Zi t )| 2 =E E K X ( Zi t -Zt )| Zi t 2 ≤ L 2 X E E Zi t -Zt 2 1 | Zi t ≤ 4L 2 X E( Zt 2 1 )
,

we obtain E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2   ≤ N -1 N 2 4L 2 X N -1 E( Zt 2 1 ) + 4L 2 X N 2 E( Zt 2 1 ) ≤ 8L 2 X N E( Zt 2 1 ), (4.1.9)
and nally

E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t )   ≤ 8L 2 X N E( Zt 2 1 ) 1/2
.

Similarly, we have

E   1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t )   ≤ 8L 2 C N E( Zt 2 1
)

1/2 . Finally, dE(r i t ) ≤ (1 + γ + 2L X + 2L C )E(r i t ) + 8L 2 X + 8L 2 C 1 N E( Zt 2 1 )
1/2 dt.

Preliminaries 143

Then by Proposition 4.1.6, there is a constant C 0 depending on initial con-

ditions such that dE(r i t ) ≤   (1 + γ + 2L X + 2L C )E(r i t ) + 8L 2 X + 8L 2 C C 0,1 √ N e 1 2 C 0,2 t   dt Conclusion : We have thus obtained d E(r i t ) + 8(L 2 X + L 2 C )C 0,1 N 1 1 + γ + 2L X + 2L C - C 0,2 2 e 1 2 C 0,2 t ≤ (1 + γ + 2L X + 2L C ) × E(r i t ) 8(L 2 X + L 2 C )C 0,1 N 1 1 + γ + 2L X + 2L C - C 0,2 2
e 1 2 C 0,2 t dt and Gronwall's lemma yields

E(r i t ) + 8(L 2 X + L 2 C )C 0,1 N 1 1 + γ + 2L X + 2L C - C 0,2 2 e 1 2 C 0,2 t ≤ e (1+γ+2L X +2L C )t E(r i 0 ) + 8(L 2 X + L 2 C )C 0,1 N 1 1 + γ + 2L X + 2L C - C 0,2 2 , thus E(r i t ) ≤ C 1 e C 2 t 1 √ N .
Let µ 0 a measure on R 2 , µ k,N t the marginal distribution at time t of the rst k neurons Z 1 t , . . . , Z k t of an N particles system (4.1.1) with initial distribution (µ 0 ) ⊗N , and μt is a solution of (4.1.2) with initial distribution µ 0 .

We obtain for Wasserstein 1 distance

W 1 (µ k,N t , μ⊗k t ) = inf E[ Z (k) -Z(k) 1 ], P Z (k) = µ k,N t , P Z(k) = μ⊗k t ≤ inf E k i=1 r i t , P (Z i,N t ) i = µ k,N t , P ( Zi t ) i = μ⊗k t ≤kE(r 1 t ) ≤C 1 e C 2 t k √ N .
Hence Theorem 4.1.3.

Preliminaries

Index Throughout this article, we dene many parameters and constants. For the sake of clarity, we list the main ones here so as to give the reader an index to refer to.

X, C, Z : X and C are the processes we consider (see (4.1.1) and (4.1.2)) and we often refer to Z = (X, C), μt = Law( Zt ) : the density of the non linear limit (see (4.1.2)), α, β, γ, σ X , σ C : parameters of the problem (see (4.1.1)), φ rc and φ sc are two Lipschitz continuous functions used to dene the coupling method, and their denitions involve a parameter ξ which converges to 0 in the end (see the beginning of Section 4.3).

K X , K C , L X , L C , L X,max , L C,max : K X (resp. K C ) is an Lipschitz con- tinuous interaction kernel, with Lipschitz constant L X ∈ [0, L X,max ] (resp. L C ∈ [0, L C,max ]),

Notations

For h : R 2N → R, for all (z i )

1≤i≤N = (x i , c i ) 1≤i≤N ∈ R 2N , the generator of (4.1.1) is L N : C 2 (R 2N , R) → C 0 (R 2N , R), given by L N h(z 1 , .., z N ) = N i=1 L i,N h,
i ≤ N , L i,N : C 2 (R 2N , R) → C 0 (R 2N , R) is dened as L i,N h(z 1 , .., z N ) =   x i -x 3 i -c i -α + 1 N N j=1 K X (z i -z j )   ∂ x i h +   γx i -c i -β + 1 N N j=1 K C (z i -z j )   ∂ c i h + σ 2 x 2 ∂ 2 x i ,x i h + σ 2 c 2 ∂ 2 c i ,c i h.
For h : R 2 → R, for all z ∈ R 2 , the generator of (4.1.2) for a given distribution µ is denoted by

L µ : C 2 (R 2N , R) → C 0 (R 2N , R) : L µ h(x, c) = x -x 3 -c -α + K X * µ(z) ∂ x h + (γx -c -β + K C * µ(z)) ∂ c h + σ 2 x 2 ∂ 2 xx h + σ 2 c 2 ∂ 2 cc h.
Remark 4.2.1. In particular, we notice that for xed (z i ) 1≤i≤N ∈ (R 2 ) N , if we consider the empirical measure µ emp = 1 N j δ z j , we have for all h : R 2 → R and z ∈ R 2 ,

L µemp h(z) =   x -x3 -c -α + 1 N N j=1 K X (z -z j )   ∂ x h +   γ x -c -β + 1 N N j=1 K C (z -z j )   ∂ c h + σ 2 x 2 ∂ 2 xx h + σ 2 c 2 ∂ 2 cc h.
In this case, if we consider z = z i for a specic i and we denote hi : (z 1 , .., z N ) → h(z i ), then L µemp h(z i ) = L i,N hi (z 1 , . . . , z N ).

First Lyapunov function

We dene a rst Lyapunov function H. It won't be sucient to control the dynamic in order to have uniform in time propagation of chaos, but it gives us a control on second moment of (Z i,N t ) i and Zt with Propositions 4.1.5 and 4.1.6. Then, we will dene in Subsection 4.2.3 a modied Lyapunov function.

Let H : R 2 → R be dened by

H(z) = H(x, c) = 1 2 γx 2 + βx + 1 2 c 2 + αc + H 0 , (4.2.2)
with

H 0 = β 2 γ + α 2 ,
where γ, β and α are the parameters of the system (4.1.1). H(z) can also be written

H(z) = γ 2 x + β γ 2 + 1 2 (c + α)
L µ H(z) ≤B + (α X L X + β X L C ) E µ (|X|) 2 -x2 + (α C L X + β C L C ) E µ (|C|) 2 -c2 -λH(z). (4.2.4)
Moreover, for all (z i ) 1≤i≤N ∈ R 2N , by denoting H : (z 1 , . . . , z N ) → H(z i ),

L i,N H(z 1 , . . . , z N ) ≤B + (α X L X + β X L C )     1 N N j=1 |x j |   2 -x 2 i   + (α C L X + β C L C )     1 N N j=1 |c j |   2 -c 2 i   -λH(z i ), (4.2.5)
with

α X = γ 2 + 1 2 , β X = 17 2 , α C = 1 16 , β C = 1 2 + 1 32
We refer to Appendix 4.A.2 for the proof of this lemma and of the following Proposition. Likewise, there are two main consequences to (4.2.5) that we will use :

d dt 1 N N i=1 EH Z i,N t ≤ B -λ 1 N N i=1 EH Z i,N t , (4.2.7)
and

L N 1 N N i=1 H Z i,N t ≤ B -λ 1 N N i=1 H Z i,N t , ( 4
L X 8 + L C 2 + 3 32 < 1, if E(| X0 | 2 ) < +∞ and E(| C0 | 2 )
< +∞, then there exists C 0 such that for all t ≥ 0:

E | Xi t | 2 + | Ci t | 2 ≤ C 0 .
From now on, we consider λ > 0 satisfying (4.2.3) (and use the a priori bounds L X,max and L C,max to ensure the existence of such a λ).

Modication of the function

Let C0 > 0, ã > 0 and consider an initial measure µ 0 on R 2 which satises E µ 0 e ã(|X|+|C|) ≤ C0 .

For technical reasons, we need a greater restoring force by the Lyapunov function than the one given in Lemma 4.2.3. We will need better control of the noise and interaction, in order to have uniform in time propagation of chaos. We thus modify it in order to obtain estimates such as ( 4

L µ H = exp a √ H L µ H + 1 2 a 2 √ H exp a √ H |σ x ∂ x H| 2 + |σ c ∂ c H| 2 = exp a √ H L µ H + a 4 √ H exp a √ H σ 2 x (γx + β) 2 + σ 2 c (c + α) 2 ≤ exp a √ H B + (α X L X + β X L C ) E µ (|X|) 2 + (α C L X + β C L C ) E µ (|C|) 2 -λH + 1 2 max σ 2 x , σ 2 c max (γ, 1) a √ H exp a √ H ≤ exp a √ H B + 1 2 max σ 2 x , σ 2 c max (γ, 1) 2 a 2 2λ + (α X L X + β X L C ) E µ (|X|) 2 + (α C L X + β C L C ) E µ (|C|) 2 - λ 2 H , (4.2.13)
where for this last inequality we used Young's inequality

1 2 max σ 2 x , σ 2 c max (γ, 1) a √ H ≤ λ 2 H + 1 2 max σ 2 x , σ 2 c max (γ,
, v i ∈ R d , L N H (z i ) ≤ exp a H (z i )   B + (α X L X + β X L C )   1 N N j=1 |x j |   2 + (α C L X + β C L C )   1 N N j=1 |c j |   2 - λ 2 H (z i )   .
Summing over i ∈ {1, .., N }, we may calculate

(α X L X + β X L C ) N j=1 N j=1 |x j | N 2 N i=1 exp a H (z i ) N - λ 16 N i=1 H (z i ) exp a H (z i ) N ≤ λ 16 N i,j=1 H (z i ) N exp a H (z j ) N - N i=1 H (z i ) exp a H (z i ) N ≤0.
(4.2.17)

Here, we used Lemma 4.2.2, the fact that ∀x, y ≥ 0 xe

√ y + ye √ x -xe √ x -ye √ y = (e √
x -e √ y )(y -x) ≤ 0 and assumed

(α X L X + β X L C ) ≤ γλ 64 .
Likewise,

(α C L X + β C L C ) N j=1 N j=1 |c j | N 2 N i=1 exp a H (z i ) N - λ 16 N i=1 H (z i ) exp a H (z i ) N ≤ 0, ( 4
(α C L X + β C L C ) ≤ λ 64 .
There is therefore a constant, which for the sake of clarity we will also denote B (as we may take the maximum of the previous constants), such that we get

L i,N H(Z i,N t ) ≤ B + (α X L X + β X L C ) N j=1 |X j,N t | N 2 exp a H Z i,N t + (α C L X + β C L C ) N j=1 |C j,N t | N 2 exp a H Z i,N t - λ 4 H Z i,N t exp a H Z i,N t (4.2.19) L N 1 N N i=1 H(Z i,N t ) ≤ B - λ 4 1 N N i=1 H(Z i,N t ) exp a H Z i,N t (4.2.20)
and

L N 1 N N i=1 H(Z i,N t ) ≤ B - λ 4 1 N N i=1 H(Z i,N t ) (4

Parameters

We start by xing the values of some parameters. The somewhat intricate expressions in this section are dictated by the computations arising in the proofs later on. They are somewhat roughly chosen and far from optimal as we only wish to convey the fact that every constant is explicit.

Recall α X , β X , α C and β C given in Lemma 4.2.3. a > 0 is xed from the last Subsection and the denition of H and B are obtained from the same Subsection. Let λ > 0 be a xed parameter satisfying

L X,max 8 + L C,max 2 + 3 32 < 1 - λ 2 .
Remark 4.2.7. In fact, to have the existence of such a λ, we have to assume that L X,max 8 + L C,max 2 + 3 32 < 1.

For example, we can consider L X,max = 4 and L C,max = 1 5 .

We consider the following parameters

δ =(1 + δ) 1 + L X,max 1 -L C,max , R 0 = 1024 B 15λ min(γ, 1) , R = 1 + δ 2 R 0 , C (f ) 1 =16γ 1 a 2 1 + a β + α δ 2 max (γ, 1) exp a 2 2 -1 + 2 max (γ, 1) √ γ + 1 δ (e -2) , C (f ) 2 =4γ 1 + a β + α δ + 2a 2 √ γ + 1 δ 2 max (γ, 1) , c = min 2 B η , λ 32
η -15 η , 1 1 + η min σ x 2 √ πR , δ(1 -L C,max ) 2(1 + δ) × exp - 1 4σ 2 x 1 + δγ + L X,max + δL C,max + C (f ) 1 + C (f ) 2 σ 2 x R 2 , = ηc 2 B , φ min = exp - 1 4σ 2 x (1 + δγ + L X,max + δL C,max + ( C (f ) 1 + C (f ) 2 )σ 2 x )R 2 , C 1 = 1 min (δ, 1) 2 φ min min 16 (1 + δ 2 ) min (γ, 1) , 1 , C 2 = 1 min (δ 2 , 1) 2 φ min min 16 (1 + δ 2 ) min (γ, 1) , 1 , C z = 2 φ min min 1, 4 max 1 γ , 1 .
We dene f as follows

f (r) = r∧R 0 φ(s)g(s)ds, (4.2.22) φ(r) = exp - 1 4σ 2 x 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x r 2 , Φ(s) = s 0 φ(u)du, g(r) =1 - c + 2 B σ 2 x r 0 Φ(s)φ(s) -1 ds.
Assume furthermore that L X and L C , the Lipschitz constants, satisfy

L X ≤ min λ 128C z , λa 512 C z , c 2C
α X L X + β X L C ≤ γλ 128 and α C L X + β C L C ≤ λ 128 .
Obviously, these constants can be optimized.

We can dene C K X ,K C of the Theorem 4.1.4 by:

C K X ,K C ≤ 1 max(δ, 1) min λ 128C z , λa 512 C z , c 2C 1 .
(

f is C 2 on (0, R) such that f + (0) = 1 and f -(R) > 0, and constant on [R, ∞).
Moreover, we assume f is non-negative, non-decreasing and concave, and for all s ≥ 0,

min (s, R) f -(R) ≤ f (s) ≤ min (s, f (R)) ≤ min (s, R) .
For all r ∈ [0, R], φ(r) ≥ φ min and g(r) ≥ 1 2 .

We have the conditions

2f (R) ≥ exp - 1 4σ 2 x 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x R 2 , 2c + 4 B ≤ 1 -L C - 1 + L X δ min r∈]0,R] f (r)r f (r) , c ≤ λ 32 64 B 15λ 1 + 64 B 15λ , 1 + L X 1 -L C < δ and ≤ 1.
The proof of this lemma is done in Appendix 4.A.3.

Intuitively, ϕ rc represents the region of space in which we consider a reection coupling (rc), and ϕ sc the one in which we consider a synchronous coupling (sc). In fact, we will dene

dB i,X t =ϕ sc |X i,N t -Xi t | dB i,sc,X t + ϕ rc |X i,N t -Xi t | dB i,rc,X t d Bi,X t =ϕ sc |X i,N t -Xi t | dB i,sc,X t -ϕ rc |X i,N t -Xi t | dB i,rc,X t .
In particular, we get d

(B i,X t -Bi,X t ) = 2ϕ rc |X i,N t -Xi t | dB i,rc,X t which will be null if x / ∈ ξ 2 , R + ξ . The coupling is then symmetric. On [ξ, R], the coupling is antithetic, with d(B i,X t -Bi,X t ) = 2dB i,rc,X t .
We thus simultaneously construct the following solutions

     dX i,N t = (X i,N t -(X i,N t ) 3 -C i,N t -α)dt + 1 N N j=1 K X (Z i t -Z j t )dt +σ x ϕ sc |X i,N t -Xi t | dB i,sc,X t + σ x ϕ rc |X i,N t -Xi t | dB i,rc,X t dC i,N t = (γX i,N t -C i,N t + β)dt + 1 N N j=1 K C (Z i t -Z j t )dt + σ c dB i,C t , and      d Xi t = ( Xi t -( Xi t ) 3 -Ci t -α)dt + K X * μt ( Zi t )dt +σ x ϕ sc |X i,N t -Xi t | dB i,sc,X t -σ x ϕ rc |X i,N t -Xi t | dB i,rc,X t d Ci t = (γ Xi t -Ci t + β)dt + K C * μt ( Zi t )dt + σ c dB i,C
t . Notice that we consider a symmetric coupling on the dynamics of C.

In fact, note that the reection is coordinate by coordinate, and isn't a global coupling as in [START_REF] Lindvall | Coupling of Multidimensional Diffusions by Reection[END_REF].

Main proof and results

Proposition 4.3.1. We denote r i t = r(Z i,N t , Zi t ) and G i t = G i ((Z j,N t ) j , ( Zj t ) j ), where G i is dened in (4.2.27). For all c ∈ R, for each i ∈ {1, . . . , N }, we have

d(e ct f (r i t )G i t ) ≤ e ct K i t dt + dM i t , (4.3.1)
where M i t is a continuous local martingale and K i t can be written as

K i t = Ki t + I 1,i t + I 2,i t + I 3,i t . (4.3.2)
We dene Ki t , I 1,i t , I 2,i t , I 3,i t and I 4,i t as followed:

Ki t =G i t 2cf (r i t ) + f (r i t ) 2σ 2 x ϕ rc |X i,N t -Xi t | 2 + f (r i t ) (1 + γδ + L X + δL C )|X i,N t -Xi t | -|(X i,N t ) 3 -( Xi t ) 3 | + (1 + L X + δL C -δ)|C i,N t -Ci t | + C (f ) 1 + C (f ) 2 σ 2 x ϕ rc |X i,N t -Xi t | 2 r i t + f (r i t )   4 B - λ 16
H( Zi t ) - λ 16
H(Z i,N t ) - λ 16N N j=1 H( Zj t ) - λ 16N N j=1 H(Z j,N t )   , (4.3.3) 4.3. Proof of Theorem 4.1.4 in the case σ X > 0 155 I 1,i t =G i t f (r i t )   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t )   + δG i t f (r i t )   1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t )   , (4.3.4) I 2,i t =G i t f (r i t )   L X N   N j=1 Z j,N t -Zj t 1     + δG i t f (r i t )   L C N   N j=1 Z j,N t -Zj t 1     -cf (r i t )G i t -f (r i t ) λ 16 H( Zi t ) exp a H( Zi t ) + λ 16 H(Z i,N t ) exp a H(Z i,N t ) + λ 16N N j=1 H( Zj t ) exp a H( Zj t ) + λ 16N N j=1 H(Z j,N t ) exp a H(Z j,N t )   , (4.3.5)
and

I 3,i t = f (r i t )   (α X L X + β X L C ) N j=1 |X j,N t | N 2 exp a H(Z i,N t ) + (α C L X + β C L C ) N j=1 |C j,N t | N 2 exp a H(Z i,N t ) - λ 16
H(Z i,N t ) exp a H(Z i,N t ) - λ 16N N j=1 H(Z j,N t ) exp a H(Z j,N t )   . (4.3.6)
We need a control on E(G i t ), which is a consequence of Lyapunov's properties on H and the initial assumption of the Theorem 4.1.4. A proof is given in Appendix 4.3.7. Lemma 4.3.2. There exists C G,1 and C G,2 , such that for each i ≤ N , for all t > 0,

we have E(G i t ) ≤ C G,1 , E[(G i t ) 2 ] ≤ C G,2 .
Each of the terms given in Proposition 4.3.1 will be controlled by dierent ways.

The following Lemmas summarizes it. The rst term, Ki t , contains the various be- haviors we have previously identied : we deal with it either through a synchronous coupling (when the deterministic drift is contracting), or through a reection coupling (notice the second derivative f " which will provide contraction provided f is suciently concave). Finally, notice the eect of Lyapunov function H which yields a restoring force.

Lemma 4.3.3. For each i ≤ N , for all t > 0,

E Ki t ≤ ξ 2 + δγ + L X + δL C -L C - 1 + L X δ EG i t . (4.3.7)
The interaction term

1 N K X (Z j,N t -Z i,N t ) -K X * μt ( Zi t ) can be decomposed into { 1 N K X ( Zj t -Zi t ) -K X * μt ( Zi t )} and 1 N K X (Z j,N t -Z i,N t ) -K X ( Zj t -Zi t )
. The rst part is in I 1,i t is dealt with using some form of law of large number in a similar way as what has been done in the proof of Theorem 4.1.3. Lemma 4.3.4. For each i ≤ N , for all t > 0, Finally, I 3,i t deals with the non linearity appearing in the dynamics of the Lya- punov function, and will be non positive for values of L X and L C suciently small.

E(I 1,i t ) ≤ 3 N -1 C 1/2 G,2 2C 0 (L X + L C ), (4.3
It is also here we justify adding the last two terms in (4.2.27).

Lemma 4.3.6. For each i ≤ N , for all t > 0, I 3,i t ≤ 0.

EK i t = 1 N N i=1 E Ki t + 1 N N i=1 EI 1,i t + 1 N N i=1 EI 2,i t + 1 N N i=1 EI 3,i t ≤ 1 N N i=1 ξ 2 + δγ + L X + δL C -L C - 1 + L X δ EG i t + 1 N N i=1 3 N -1 C 1/2 G,2 2C 0 (L X + L C ) ≤ξ 2 + δγ + L X + δL C -L C - 1 + L X δ 1 N N i=1 EG i t + 3 N -1 C 1/2 G,2 2C 0 (L X + L C )
Since by Lemma 4.3.2, we have

1 N N i=1 EG i t ≤C G,1 , we obtain 1 N N i=1 EK i t ≤ξA + (L X + L C ) B √ N
where A and B are constants. For all initial couplings such that Eρ (Z j 0 , Zj 0 ) 1≤j≤N < ∞, by taking the expectation of (4.3.1) along a sequence of increasing stopping times, we have thanks to Fatou's lemma

e ct E ρ (Z j t , Zj t ) 1≤j≤N ≤E ρ (Z j 0 , Zj 0 ) 1≤j≤N + ξA t 0 e cs ds + (L X + L C ) B √ N t 0 e cs ds ≤E ρ (Z j 0 , Zj 0 ) 1≤j≤N + ξA e ct -1 c + (L X + L C ) B √ N e ct -1 c .
We obtain

E ρ (Z j t , Zj t ) 1≤j≤N ≤E ρ (Z j 0 , Zj 0 ) 1≤j≤N e -ct + ξA c 1 -e -ct + (L X + L C )B c 1 √ N 1 -e -ct .
By using the exchangeability of the particles, we have

E ρ (Z j t , Zj t ) 1≤j≤N = E 1 N N i=1 f (r i t )G i t = E 1 k k i=1 f (r i t )G i t for all k ∈ N. Then E k i=1 f (r i t )G i t = kE ρ (Z j t , Zj t ) 1≤j≤N .
Let µ 0 a measure on R 2 , µ k,N t the marginal distribution at time t of the rst k neurons (X 1 t , C 1 t ), ...., (X k t , C k t ) of an N particles system (4.1.1) with initial distribution (µ 0 ) ⊗N , and μt is a solution of (4.1.2) with initial distribution µ 0 . This implies E ρ (Z j 0 , Zj 0 ) 1≤j≤N = 0. By Lemma 4.2.10, we obtain for Wasserstein 1 distance

W 1 (µ k,N t , μ⊗k t ) = inf E[ Z (k) -Z(k) 1 ], P Z (k) = µ k,N t , P Z(k) = μ⊗k t = inf E k i=1 Z i,N t -Zi t 1 , P (Z i,N t ) i = µ k,N t , P ( Zi t ) i = μ⊗k t ≤ inf C 1 E k i=1 f (r i t )G i t , P (Z i,N t ) i = µ k,N t , P ( Zi t ) i = μ⊗k t ≤ inf kC 1 E ρ (Z j t , Zj t ) 1≤j≤N , P (Z i,N t ) i = µ k,N t , P ( Zi t ) i = μ⊗k t ≤kC 1 inf E ρ (Z j 0 , Zj 0 ) 1≤j≤N + (L X + L C )B c 1 √ N , P (Z i,N t ) i = µ k,N t , P ( Zi t ) i = μ⊗k t ≤ ξAkC 1 c 1 -e -ct + (L X + L C )BC 1 c k √ N 1 -e -ct
By taking the limit as ξ → 0 uniformly in time, we obtain the desired result. The same lemma and the same type of calculations yield the result for Wasserstein 2

W 2 (µ k,N t , μ⊗k t ) 2 ≤ k √ N C 2 (L X + L C )B c .

Proof of the decomposition

Proof of Proposition 4.3.1. First, we need to calculate d(e ct f (r i t )G i t ), where we recall

r i t = |X i,N t -Xi t | + δ|C i,N t -Ci t | and G i t = 1 + H( Zi t ) + H(Z i,N t ) + N N j=1 H(Z j,N t ) + N N j=1 H( Zj t ).
We have already calculated d(X

d|X i,N t -Xi t | =A X t dt + 2sign(X i,N t -Xi t )σ x ϕ rc |X i,N t -Xi t | dB i,rc,X t , with A X t ≤|X i,N t -Xi t | -(X i,N t ) 3 -( Xi t ) 3 + C i,N t -Ci t + 1 N N j=1 K X (Z i t -Z j t ) -K X * μt ( Zi t ) .
Likewise, as it has already been calculated in (4.1.8) in Subsection 4.1.4,

d|C i,N t -Ci t | = A C t dt, (4.3.11) with A C t ≤ γ X i,N t -Xi t -|C i,N t -Ci t | + 1 N N j=1 K C (Z i t -Z j t ) -K C * μt ( Zi t ) . Now we have dr i t = A X t + δA C t dt + 2sign(X i,N t -Xi t )σ x ϕ rc |X i,N t -Xi t | dB i,rc,X t
and we deduce with the Ito's formula

df (r i t ) = f (r i t )dr i t + 1 2 f (r i t ) 2σ x ϕ rc |X i,N t -Xi t | 2 dt.
Finally, for c > 0,

d(e ct f (r i t )) = ce ct f (r i t )dt + e ct df (r i t ).
Then, by Ito's formula,

1 dG i t = L μt H( Zi t ) + L N H(Z i,N t ) dt + σ x ϕ rc |X i,N t -Xi t | ∂ x H(Z i,N t ) -∂ x H( Zi t ) dB i,rc,X t + σ x ϕ sc |X i,N t -Xi t | ∂ x H(Z i,N t ) + ∂ x H( Zi t ) dB i,sc,X t + σ c ∂ c H(Z i,N t ) + ∂ c H( Zi t ) dB i,C t + 1 N N j=1 L μt H( Zj t ) + L N H(Z j,N t ) dt + σ x N N j=1 ϕ rc |X j,N t -Xj t | ∂ x H(Z j,N t ) -∂ x H( Zj t ) dB j,rc,X t + σ x N N j=1 ϕ sc |X j,N t -Xj t | ∂ x H(Z j,N t ) + ∂ x H( Zj t ) dB j,sc,X t + σ c N N j=1 ∂ c H(Z j,N t ) + ∂ c H( Zj t ) dB j,C t .
We nally get

d(e ct f (r i t )G i t ) =G i t d(e ct f (r i t )) + e ct f (r i t )dG i t + 2 1 + 1 N σ 2 x ϕ rc |X i,N t -Xi t | 2 × sign(X i,N t -Xi t ) ∂ x H(Z i,N t ) -∂ x H( Zi t ) e ct f (r i t )dt.
Now, we need to use the following Lemma, proven in Appendix 4.A.5, to have a more tractable expression: Lemma 4.3.7. We have the majoration

2 1 + 1 N σ 2 x ϕ rc |X i,N t -Xi t | 2 sign(X i,N t -Xi t ) ∂ x H(Z i,N t ) -∂ x H( Zi t ) ≤ C (f ) 1 + C (f ) 2 σ 2 x ϕ rc |X i,N t -Xi t | 2 r i t G i t .
Eventually, by denoting the terms in dB i,rc,X t , dB i,sc,X t , dB i,C t , ... as the local martingale dM i t , we obtain

d(e ct f (r i t )G i t ) ≤G i t ce ct f (r i t )dt + e ct G i t f (r i t ) A X t + δA C t dt + e ct G i t 1 2 f (r i t ) 2σ x ϕ rc |X i,N t -Xi t | 2 dt + e ct f (r i t ) L μt H( Zi t ) + L N H(Z i,N t ) dt + e ct f (r i t ) 1 N N j=1 L μt H( Zj t ) + L N H(Z j,N t ) dt + C (f ) 1 + C (f ) 2 σ 2 x ϕ rc |X i,N t -Xi t | 2 r i t G i t e ct f (r i t )dt + dM i t .
We

I 1,i t =G i t f (r i t )   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t )   + δG i t f (r i t )   1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t )   .
The second part of the decomposition is grouped in I

I 2,i t =G i t f (r i t )   L X N   N j=1 |X j,N t -Xj t | + |C j,N t -Cj t |     + δG i t f (r i t )   L C N   N j=1 |X j,N t -Xj t | + |C j,N t -Cj t |     -cf (r i t )G i t -f (r i t ) λ 16 H( Zi t ) exp a H( Zi t ) + λ 16 H(Z i,N t ) exp a H(Z i,N t ) + λ 16N N j=1 H( Zj t ) exp a H( Zj t ) + λ 16N N j=1 H(Z j,N t ) exp a H(Z j,N t )   .
We gather the expectations terms, obtained with (4.2.19), in I 3,i t , and we keep a fraction of Lyapunov function to control it:

I 3,i t = f (r i t )   (α X L X + β X L C ) N j=1 |X j,N t | N 2 exp a H(Z i,N t ) + (α C L X + β C L C ) N j=1 |C j,N t | N 2 exp a H(Z i,N t ) - λ 16
H(Z i,N t ) exp a H(Z i,N t ) - λ 16N N j=1 H(Z j,N t ) exp a H(Z j,N t )   .
Finally, we dene Ki t with the leftovers. It will, in particular, give the constraints on f which explain its choice.

Ki

t =G i t 2cf (r i t ) + f (r i t ) 2σ 2 x ϕ rc |X i,N t -Xi t | 2 + f (r i t ) (1 + γδ + L X + δL C )|X i,N t -Xi t | -|(X i,N t ) 3 -( Xi t ) 3 | + (1 + L X + δL C -δ)|C i,N t -Ci t | + C (f ) 1 + C (f ) 2 σ 2 x ϕ rc |X i,N t -Xi t | 2 r i t + f (r i t )   4 B - λ 16
H( Zi t ) - λ 16
H(Z i,N t ) - λ 16N N j=1 H( Zj t ) - λ 16N N j=1 H(Z j,N t )   , 162
4 γ (α X L X + β X L C ) ≤ λ 32 and 4 (α C L X + β C L C ) ≤ λ 32 ,
and since

H(Z j,N t ) exp a H(Z i,N t ) ≤ H(Z i,N t ) exp a H(Z i,N t ) + H(Z j,N t ) exp a H(Z j,N t ) we obtain (α X L X + β X L C ) N j=1 |X j,N t | N 2 exp a H(Z i,N t ) + (α C L X + β C L C ) N j=1 |C j,N t | N 2 exp a H(Z i,N t ) - λ 16N   N H(Z i,N t ) exp a H(Z i,N t ) + N j=1 H(Z j,N t ) exp a H(Z j,N t )   ≤ 0.
Then, for each i ≤ N , and for all t > 0, I 3,i t ≤ 0.

Proof of Lemma 4.3.5. We prove the non-positivity of 1

N N i=1 I 2,i
t , where we recall for each 1 ≤ i ≤ N , for all t > 0

I 2,i t =G i t f (r i t )   L X N   N j=1 |X j,N t -Xj t | + |C j,N t -Cj t |     + δG i t f (r i t )   L C N   N j=1 |X j,N t -Xj t | + |C j,N t -Cj t |     -cf (r i t )G i t -f (r i t ) λ 16 H( Zi t ) exp a H( Zi t ) + λ 16 H(Z i,N t ) exp a H(Z i,N t ) + λ 16N N j=1 H( Zj t ) exp a H( Zj t ) + λ 16N N j=1 H(Z j,N t ) exp a H(Z j,N t )   . First, note N i=1 G i t =N + N i=1 H Zi t + H(Z i,N t ) + N N i=1 N k=1 H Zk t + H(Z k,N t ) =N + 2 N i=1 H Zi t + H(Z i,N t ) . Since f r i t ≤ 1, we have 1 N N i=1   1 N f r i t G i t N j=1 Z j,N t -Zj t 1   ≤ 1 N 2   N j=1 Z j,N t -Zj t 1   N i=1 G i t ≤ 1 N N i=1 Z i,N t -Zi t 1 + 2 N 2 N i,j=1 Z i,N t -Zi t 1 H Zj t + H(Z j,N t )
and, using Lemma 4.2.10 (i)

1 N N i=1 Z i,N t -Zi t 1 ≤ C 1 N N i=1 f (r i t )G i t and with Lemma 4.2.10 (iii) N i,j=1 Z i,N t -Zi t 1 H Zj t + H(Z j,N t ) ≤C z N i,j=1 f (r i t ) 1 + H(Z i,N t ) + H( Zi t ) H Zj t + H(Z j,N t ) ≤C z N i,j=1 f (r i t ) H Zj t + H(Z j,N t ) + C z N i,j=1 f (r i t ) H(Z i,N t ) + H( Zi t ) H Zj t + H(Z j,N t ) .
Using (4.2.10) from Lemma 4.2.6, we obtain for the rst sum:

C z N i,j=1 f (r i t ) H Zj t + H(Z j,N t ) ≤ C z N i,j=1 f (r i t ) H( Zj t ) exp a H( Zj t ) + H(Z j,N t ) exp a H(Z j,N t ) .
With (4.2.11) from the same Lemma, we obtain for the second sum:

C z N i,j=1 f (r i t ) H(Z i,N t ) + H( Zi t ) H Zj t + H(Z j,N t ) ≤ C z 2 a N i,j=1 f (r i t ) H(Z i,N t ) + H( Zi t ) H Zj t exp a H Zj t + H Z j,N t exp a H Z j,N t .
Since for all (y 1 , y 2 , y 3 , y 4 ) ∈ (R + )

2 C z a N i,j=1 f (r i t ) H(Z i,N t ) + H( Zi t ) H Zj t exp a H Zj t + H Z j,N t exp a H Z j,N t ≤ 4 C z a N i,j=1 f (r i t ) H( Zi t ) exp a H( Zi t ) + H(Z i,N t ) exp a H(Z i,N t ) + 4 C z a N i,j=1 f (r i t ) H( Zj t ) exp a H( Zj t ) + H(Z j,N t ) exp a H(Z j,N t ) ≤ 4 C z a N N i=1 f (r i t ) H( Zi t ) exp a H( Zi t ) + H(Z i,N t ) exp a H(Z i,N t ) + 4 C z a N i,j=1 f (r i t ) H( Zj t ) exp a H( Zj t ) + H(Z j,N t ) exp a H(Z j,N t ) .
Then, by reconsidering the rst expression:

1 N N i=1   1 N f r i t G i t N j=1 Z j,N t -Zj t 1   ≤ C 1 N N i=1 f (r i t )G i t + 2 N 2 C z N i,j=1 f (r i t ) H( Zj t ) exp a H( Zj t ) + H(Z j,N t ) exp a H(Z j,N t ) + 2 N 2 4 C z a N N i=1 f (r i t ) H( Zi t ) exp a H( Zi t ) + H(Z i,N t ) exp a H(Z i,N t ) + 2 N 2 4 C z a N i,j=1 f (r i t ) H( Zj t ) exp a H( Zj t ) + H(Z j,N t ) exp a H(Z j,N t )
This way, by Assumption 4.2.8 since

L X C 1 ≤ c 2 , 2C z L X ≤ λ 64 and L X 8C z a ≤ λ 64 , we get 1 N N i=1   1 N f r i t G i t N j=1 Z j,N t -Zj t 1   ≤ 1 N c 2L X N i=1 f (r i t )G i t + N 2 λ 64L X N i,j=1 f (r i t ) H( Zj t ) exp a H( Zj t ) + H(Z j,N t ) exp a H(Z j,N t ) + N λ 64L X N i=1 f (r i t ) H( Zi t ) exp a H( Zi t ) + H(Z i,N t ) exp a H(Z i,N t ) + N 2 λ 64L X N i,j=1 f (r i t ) H( Zj t ) exp a H( Zj t ) + H(Z j,N t ) exp a H(Z j,N t )
and we nally obtain "half" the result we obtain the second "half"

1 N N i=1 G i t f (r i t )   L X N   N j=1 Z j,N t -Zj t 1     - c 2 1 N N i=1 f (r i t )G i t - 2 1 N N i=1 f (r i t ) λ 16
1 N N i=1 δG i t f (r i t )   L C N   N j=1 Z j,N t -Zj t 1     - c 2 1 N N i=1 f (r i t )G i t - 2 1 N N i=1 f (r i t ) λ 16 H( Zi t ) exp a H( Zi t ) + λ 16 H(Z i,N t ) exp a H(Z i,N t ) λ 16N N j=1 H( Zj t ) exp a H( Zj t ) + λ 16N N j=1 H(Z j,N t ) exp a H(Z j,N t )   ≤ 0.
Eventually, we have proved

N i=1 I 2,i t ≤ 0.
Proof of Lemma 4.3.4. Since f (r) ≤ 1, we have by Cauchy-Schwarz

E   G i t f (r i t )   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t )     ≤E |G i t | 2 1/2 E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2   1/2 . By Lemma 4.3.2, we have for each i ≤ N , for all t ≥ 0, E[(G i t ) 2 ] ≤ C G,2 .
Moreover, we notice that ( Zj t ) j are i.i.d with law μt . Let's denote Zt a generic random variable of law μt independent of Zi t . The calculus of the right term of the product has already be done in Subsection 4.1.4, and we have (4.1.9):

E   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t ) 2   ≤ 8L 2 X N E( Zt 2 1 ) ≤ 8L 2 X N -1 E( Zt 2 1
).

A similar calculus gives us

E   1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t ) 2   ≤ 8L 2 C N -1 E( Zt 2 1 )
.

By Lemma 4.2.5, E | Xt | 2 + | Ct | 2 ≤ C 0 . In particular, E Zt 2 1 = E | Xt | + | Ct | 2 ≤ 2E | Xt | 2 + | Ct | 2 ≤ 2C 0 . Thus E   G i t f (r i t )   1 N N j=1 K X ( Zi t -Zj t ) -K X * μt ( Zi t )     ≤ L X C 1/2 G,2 2C 0 8 N -1 and likewise E   G i t f (r i t )   1 N N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t )     ≤ L C C 1/2 G,2 2C 0 8 N -1 .

Contraction in various regions of space

The goal of this section is to prove the Lemma 4.3.3: for each i ≤ N , for all t > 0,

E Ki t ≤ ξ 2 + δγ + L X + δL C -L C - 1 + L X δ EG i t . Recall Ki t =G i t 2cf (r i t ) + f (r i t ) 2σ 2 x ϕ rc |X i,N t -Xi t | 2 + f (r i t ) (1 + γδ + L X + δL C )|X i,N t -Xi t | -|(X i,N t ) 3 -( Xi t ) 3 | + (1 + L X + δL C -δ)|C i,N t -Ci t | + C (f ) 1 + C (f ) 2 σ 2 x ϕ rc |X i,N t -Xi t | 2 r i t + f (r i t )   4 B - λ 16
H( Zi t ) - λ 16
H(Z i,N t ) - λ 16N N j=1 H( Zj t ) - λ 16N N j=1 H(Z j,N t )   ,
which is a quantity that contains every term we have not yet dealt with. To prove Lemma 4.3.3, we divide for each i ∈ {1, ..., N } the space into three regions Reg

i 1 = ( Zi t , Z i,N t ) s.t. r i t ≤ R and | Xi t -X i,N t | ≥ ξ , Reg i 2 = ( Zi t , Z i,N t ) s.t. r i t ≤ R and | Xi t -X i,N t | < ξ , Reg i 3 = ( Zi t , Z i,N t ) s.t. r i t > R ,
and consider

1 N N i=1 E Ki t = 1 N N i=1 E Ki t 1 Reg i 1 + E Ki t 1 Reg i 2 + E Ki t 1 Reg i 3 .
This division in dierent regions isolates the region 1, where the coupling is reexive. Nevertheless, it doesn't mean the coupling is antithetic on the two others. The function f is constant on region 3, therefore terms in ϕ rc will be null. In region 2, the choice of the function f will make this term non-positive.

In fact, we will prove that for each i ≤ N , for all t > 0 Ki

t 1 Reg i 1 ≤ 0 and Ki t 1 Reg i 3 ≤ 0, E Ki t 1 Reg i 2 ≤ ξ 2 + δγ + L X + δL C -L C - 1 + L X δ E G i t . 4.3.4.1 Region 1 : ξ ≤ |X i,N t -Xi t | and r i t ≤ R.
In this region of space, since the coupling is reexive and ϕ

rc (|X i,N t -Xi t |) = 1, we have Ki t 1 Reg i 1 =G i t 2cf (r i t ) + 2σ 2 x f (r i t ) + f (r i t ) C (f ) 1 + C (f ) 2 σ 2 x r i t + f (r i t )(1 + γδ + L X + δL C )|X i,N t -Xi t | -G i t f (r i t )|(X i,N t ) 3 -( Xi t ) 3 | -G i t f (r i t )(δ -1 -L X -δL C )|C i,N t -Ci t | + f (r i t )4 B -f (r i t )   λ 16
H( Zi t ) + λ 16 H(Z i,N t ) + λ 16N N j=1 H( Zj t ) + λ 16N N j=1 H(Z j,N t )   , 168
≤G i t (2c + 4 B)f (r i t ) + 2σ 2 x f (r i t ) +f (r i t ) 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x r i t .
Using the denition f given in (4.2.22) we get

2σ 2 x f (r i t ) + f (r i t ) 1 + δγ + L X + L C + C (f ) 1 + C (f ) 2 σ 2 x r i t =2σ 2 x φ (r i t )g(r i t ) + 2σ 2 x φ(r i t )g (r i t ) + φ(r i t )g(r i t ) 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x r i t =2σ 2 x φ(r i t )g (r i t ) = -(2c + 4 B)Φ(r i t ). Thus (2c + 4 B)f (r i t ) + 2σ 2 x f (r i t ) + f (r i t ) 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x r i t = (2c + 4 B)f (r i t ) -(2c + 4 B)Φ(r i t ) (
≤G i t f (r i t )ξ 1 + γδ + L X + δL C + 1 -L C - 1 + L X δ .
Finally, since f (r) ≤ 1,

E Ki t 1 Reg i 2 ≤ ξ 2 + δγ + L X + δL C -L C - 1 + L X δ EG i t .
4.3.4.3 Region 3 :

r i t ≥ R.
In this region of space f = f = 0 and f is constant, and we therefore have First, we focus on interaction terms. We have

|K X * µ(z)| ≤ R 2 |K X (z -z )|µ(dz ) = R 2
L X ( z 1 + z 1 )µ(dz ).

Hence, The last inequality (4.2.8) simply relies on the sum of (4.2.5) for each i and the fact that L j,N H Z i,N t = 0 for i = j:

(
1 N N i=1 L N H Z i,N t = 1 N N i=1 L i,N H Z i,N t ≤ 1 N N i=1   B + (α X L X + β X L C )   1 N N k=1 |X k,N t | 2 -(X i,N t ) 2   + (α C L X + β C L C )   1 N N k=1 |C k,N t | 2 -(C i,N t ) 2   -λH Z i,N t   ≤B -λ 1 N N i=1 H Z i,N t .
The last inequality uses the fact that

1 N N i=1 |y i | 2 -1 N N i=1 (y i ) 2 ≤ 0 for all (y i ) 1≤i≤N ∈ R N .
Bounds on the second moments of processes We can now prove the uniform in bounds on the second moments of X i,N t , C i,N t , Xi t and Ci t .

Proof of Proposition 4.1.5. K X and K C are Lipschitz with constants L X and L C respectively. We do not assume any bound on these constants.

We assume for each i ≤ N , E(|X i,N 0 | 2 ) < +∞ and E(|C i,N 0 | 2 ) < +∞. Here, we also assume

L X 8 + L C 2 + 3 32 < 1.
Then, we can chose λ > 0 such that

L X 8 + L C 2 + 3 32 < 1 - λ 2 .
The last proof gives us

γ 4 E | Xi t | 2 + 1 4 E | Ci t | 2 ≤e -λt γE | Xi 0 | 2 + E | Ci 0 | 2 + 3 2 H 0 - B λ + B λ . If B λ < γE | Xi 0 | 2 + E | Ci 0 | 2 + 3 2 H 0 , then we get γ 4 E | Xi t | 2 + 1 4 E | Ci t | 2 ≤1 × γE | Xi 0 | 2 + E | Ci 0 | 2 + 3 2 H 0 - B λ + B λ ≤γE | Xi 0 | 2 + E | Ci 0 | 2 + 3 2 H 0 ,
which is an uniform in time bound.

Else we get

γ 4 E | Xi t | 2 + 1 4 E | Ci t | 2 ≤0 + B λ ,
which is also an uniform in time bound. η .

The appearance of φ (R) suggests we should try to minimize it. We search for δ such that φ (r) ≥ e -2 exp -(1 + γ) L X + δL C + 1 4σ 2

x r 2 on [0, R].

We recall φ(r) = exp -

1 4σ 2 x 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x r 2 ≥ exp - 1 4σ 2 x 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x r 2 .
It is therefore sucient for (4.A.2) to have

c ≤ 1 1 + η σ x 2 √ π 1 R exp - 1 4σ 2 x 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x R 2 ,
and for (4.A.3) to have

c ≤ 1 2(1 + η) δ (1 -L C ) 1 + δ exp - 1 4σ 2 x 1 + δγ + L X + δL C + C (f ) 1 + C (f ) 2 σ 2 x R 2 .
Finally, we bound L X and L C by either 0 or L X,max and L C,max , to obtain bounds on c independent of L X and L C . If r(z, z ) ≤ 1 ≤ R, we have, using Lemma 4.2.9 r(z, z

) ≤ f (r) f -(R) ≤ f (r) φ(R)g(R)
1 + H(z) + H(z ) .

If r(z, z ) ≥ 1, we have, using (4.A.

f (r) φ(R)g(R)
1 + H(z) + H(z ) . f (r) φ(R)g(R)

1 + H(z) + H(z ) .

If r(z, z ) ≤ 1 ≤ R, we have, using Lemma 4.2.9 r(z, z

) 2 ≤ r(z, z ) ≤ f (r) f -(R) ≤ f (r) φ(R)g(R)
1 + H(z) + H(z ) . Proof of the second control of the L 1 distance : We have, if r(z, z

) ≤ 1 ≤ R r(z, z ) ≤ f (r) f -(R) ≤ f (r) φ(R)g(R)
1 + H(z) + H(z ) . Then, E exp 2a H Zj 0 is bounded and we deduce E( H( Zj t )) is bounded for each j ≤ N and all t ≥ 0.

The same calculations can be done for Z j,N t . By (4.2.21), we have

L N 1 N N i=1 H(Z i,N t ) ≤ B - λ 4 1 N N i=1
H(Z i,N t ) .

In particular,

d dt E 1 N N i=1 H(Z i,N t ) ≤E L N 1 N N i=1 H(Z i,N t ) ≤ B - λ 4 E 1 N N i=1 H(Z i,N t ) ,
and we can use Gronwall's lemma. The following is exactly as above.

Finally, we have proved that for each j ≤ N , E( H(Z j,N t )) and E( H( Zj t )) are bounded uniformly in time . Thus, E(G i t ) is bounded uniformly in time.

To bound the second moment of G i t , we have to bound each type of the fol- lowing expectations: E[ H(Z 4.B Proof of Theorem 4.1.4 in the case σ X = 0 and σ C > 0

We quickly explain in this section how we may also deal with the case σ X = 0 and σ C > 0. Recall how the choice of the coupling method was motivated by the observation in (4.1.8) that the dierence of potentials C i,N t -Ci t was naturally contracting when X i,N t -Xi t was close to 0. This lead us to using a reection coupling on the Brownian motions acting on the potential X, to bring the dierence We dene Ki t , I On asymptotic behaviour of stochastic processes on neuroscience Abstract: In this thesis, we focus on two stochastic models which can be applied to neuroscience : Hawkes model and FitzHugh-Nagumo model. We study their long-time behavior.

The rst chapter deals with cumulative processes, which are a larger processes class than renewal processes. These processes accumulate independent random variables over time. These random variables are added on time intervals given by a renewal process. Inspired by the work of Lefevere, Mariani and Zambotti (2011), we prove a Large Deviations Principle for these processes, and large deviations inequalities in a more general framework.

The second chapter is dedicated to Hawkes processes, in a non-linear context, with a signed reproduction function. They model self-excitation and self-inhibition. We prove a law of large numbers, a central limit theorem and large deviations results for a unique Hawkes process. These results lie on a renewal structure for these processes introduced by Costa, Graham, Marsalle and Tran (2020), which leads to a comparison with cumulative processes. Thus, we use known results for cumulative processes and results obtained in Chapter 1. We also exhibit two examples with explicit computations. The last chapter is a joint work with Pierre Le Bris and is devoted to the study of stochastic FitzHugh-Nagumo processes in interaction. The specicity of this model, described by Stochastic Dierential Equations, is its cubic term in the drift which is non-Lipschitz. We focus on mean-eld interactions and we prove a propagation of chaos, non-uniform in time rst, and then a uniform in time one. To do so, we use a combined coupling method, i.e. a synchronous coupling on a specic subspace and an antithetic coupling on the complementary subspace. We also exhibit explicit bounds for these results.

Figure 1 . 1

 11 Figure 1.1 Représentation d'un neurone, par Nicolas Rougier (2007) 1 Les signaux sont transmis par des synapses. Ces synapses sont les points de jonction entre deux neurones, généralement entre un axone et une dendrite. Deux types de synapses existent : les synapses chimiques et les synapses électriques. Dans la synapse chimique, le signal nerveux est transmis par une molécule spécique appelée neurotransmetteur. Celle-ci est émise par le neurone qui envoie le signal et se lie à des récepteurs sur le second neurone. Dans la synapse électrique, l'inux nerveux se transmet par courant ionique : les ions se déplacent d'un neurone à un autre. Ces deux types de synapses ont chacune des caractéristiques propres et des modélisations distinctes.

i) ∃θ 0 ∈

 0 (0, +∞] tel que E[e θτ ] < ∞ pour tout θ < θ 0 , ii) ∃β 0 ∈ (0, +∞] tel que E[e β|W | ] < ∞, pour tout β < β 0 .Nous introduisons les transformées de Cramer classiques, pour (a, b) ∈ R 2 , Λ * (a, b) = sup x,y {ax + by -ln E(e xτ +yW )} (1.2.4) Λ * n (a, b) = sup x,y ax + by -ln E e xτ +yW n , (1.2.5) où W n est une réduction bien choisie de W . Nous introduisons nalement les fonctions de taux J n (m) = inf β>0

  Grâce à nos résultats de comparaison et à l'étude précise du cas d'annulation d'intensité (N g t pour g = -λ1 [0,A] ) obtenus dans la Remarque 3.2.8 dans le Chapitre 3, on a

  Pour conclure, le résultat principal de ce chapitre est une propagation du chaos uniforme en temps dans le modèle de FitzHugh-Nagumo, et est démontrée à partir d'un couplage mixte entre synchrone et antithétique. Nous avons conservé un modèle assez général, en envisageant un bruit et une interaction sur chacune des coordonnées. Cette méthode de couplage nous a également permis de dénir des constantes de vitesse explicites.

( 2 .

 2 4.8)

2. 5 .

 5 The functional I n 53 Now, we study η L,M k :

t 1 a+b -ν n t 1

 11 a+b> δ} are almost surely empty.

Remark 2 . 8 . 2 .

 282 The rst two items i) and ii) in the previous lemma are still hold true if we replace {1, ..., n} by a compact set W. ♦ 2.8.3 End of the proof.

  we may nd a sequence µ k ∈ ∆ n 0 such that µ k (ϕ) = m k and I n 0 (µ k ) ≤ β. The corresponding π k satises lim sup π k (1/(a + b)) < +∞ according to proposition 2.5.3 (here α k = 1). Since |ϕ(a, b, c)| ≤ C/(a + b), m k is bounded and one can nd a convergent subsequence.

  we obtain by dierentiating whether y is positive or negative that x + my -β ln E[e xτ +yWn ] ≥ x + my -β ln E[e xτ +|y||W | ]. Therefore, we deduce the lower bound J n (m) ≥ J |.| (m) := inf β>0 sup x∈R,y≥0 x + |m|y -β ln E[e xτ +y|W | ] . Remark that J |.| is an even function, thus by symmetry, we can assume m ≥ 0. Now, using Cauchy-Schwarz inequality, we deduce that sup x∈R,y≥0 x + my -β ln E[e xτ +y|W | ]

  proposition 2.5.3 tells us in addition that lim sup n µ n (1/(a + b)) = M < +∞ (since the sequence α n = 1). It followsµ(1/(a + b + ε)) = lim n µ n (1/a + b + ε)) ≤ lim sup n µ n (1/(a + b)) = Mso that again using monotone convergence we deduce µ(1/(a + b)) ≤ M . Hence, since c is µ a.s. bounded, µ(c/(a + b)) is well dened. To calculate µ(c/(a + b)) we need more. Using the denition of Jn and what precedes the sequence µ n satises H(π n |ψ n ) = m n µ n (1/(a + b)) ≤ C for some C < +∞ since m n is bounded and lim n µ n (1/(a + b)) = µ(1/(a + b)) > 0.

x

  + my -β ln E(e xτ +yW ) := inf β>0 sup x,y Λ(m, β, x, y) .

  For β > M -x τ := β τ one has sup x,y Λ(m, β, x, y) > M so that for all m ∈ {J ≤ M } it holds J(m) = inf 0<β≤βτ sup x,y Λ(m, β, x, y) . Now remark that sup x,y Λ(m, β, x, y) ≥ sup y Λ(m, β, 0, y) ≥ Λ(m, β, 0, κ)

Lemma 2 .A. 2 (

 22 Lemma 2.3 from[START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] -slightly dierent proof ). The map π → π(1/τ )H(π|ψ n ) is convex on {π ∈ M 1 ((0, +∞)×{1, ..., n}), π(1/τ ) < ∞}. Moreover I n is also convex.

<0 < 1 .

 1 +∞)×{1,...,n} e τ f (1,τ,W ) ψ n (dτ, dW ) = (0,+∞)×{1,...,n} e d1 (ε,δ] (τ )-D1 (δ,M ] (τ ) ψ n (dτ, dW ) = (0,+∞)×{1,...,n}e 0 1 τ ≤ε + e d 1 (ε,δ] (τ ) + e -D 1 (δ,M ] (τ ) + e 0 1 τ >M ψ n (dτ, dW ) = 1 + (e d -1)ψ n (τ ∈ (ε, δ]) + (e -D -1)ψ n (τ ∈ (δ, M ])and for s > 0 (s,+∞)

Proposition 2 .A. 7 (

 27 Equivalent of the proposition 3.3). For all f in Γ = {f : (0, +∞] 2 × {1, ..., n} → R, bounded and lower continuous such that C n,f < 1, D n,f < +∞}, sup t>0 E(e tµ n t (f ) ) < +∞ (2.A.6)

Lemma 2 . 7 . 3 Lemma 2 .A. 9 (

 27329 Lemma 2.7.3 -Analogue of Lemma 3.2 in [52] -Slightly dierent proof ). For f ∈ C b ((0, +∞] 2 × {1, ..., n}) we dene

  , T -us, k) -f (+∞, +∞, k)|du, E(s) := sup u≥s,k≤n ξ(u, k).

  |f (us, T -us, k) -f (+∞, +∞, k)| -→ s→∞ 0. The Fatou's lemma allows us to conclude: , T -us, k) -f (+∞, +∞, k)|du ≤ , τ -us, k) -f (+∞, +∞, k)|du = 0

Contents 3 . 1

 31 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.2 Notation, denitions and results. . . . . . . . . . . . . . . . . 106 3.2.1 Hawkes processes . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.2.2 Denitions and assumptions . . . . . . . . . . . . . . . . . . . 107 3.2.3 Hawkes processes as cumulative processes. . . . . . . . . . . . 109 3.2.4 LLN, CLT and asymptotic deviations for signed reproduction function h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.3 One more example with explicit calculations: canceling intensity with delay. . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.4 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.4.1 Proofs of Proposition 3.2.6 and 3.2.7 . . . . . . . . . . . . . . 120 3.4.2 Proof of the LLN and CLT . . . . . . . . . . . . . . . . . . . 123 3.4.3 Proof of Corollary 3.2.13 . . . . . . . . . . . . . . . . . . . . . 124 3.5 Final comments. . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.1 Introduction.

3. 2 .

 2 Notation, denitions and results. 107 Denition 3.2.1. Let λ > 0 and h : (0, +∞) → R a signed measurable function.

Figure 3 . 2

 32 Figure 3.2 Example of the evolution of intensity in function of time and renumbering of jumps in the case where h = -λ 1 [1,2] .

Figure 3 .

 3 Figure 3.2 is an example of this splitting of the time and the renumbering of the jumps, in the case where h(t) = -λ1 (1,2) (t), so that L(h) = 2.The next Proposition gathers important properties on the law of (τ i , W i ) dened

Denition 3 . 2 . 11 .

 3211 We introduce the Cramer transform for (a, b) ∈ R 2 , Λ * (a, b) = sup x,y ax + by -ln E e xτ 1 +yW 1 .

Remark 3 . 3 . 2 .

 332 This result naturally leads to some comments on the issues brought by inhibition.

2 )

 2 will conclude the LLN and (3.4.3) the CLT.

  κ 1 )aθ 0 yielding the result with κ = κ 1 κ 2 and κ = 1 -κ 1 . The condition κ + 2κ = 1 arises from the equality of the last two terms. Finally, (3.2.11) is a consequence of the same reasoning on N h t

3. 5 . 3 . 5

 535 Final comments. 127 Final comments.

4.A. 4

 4 Proof of Lemma 4.2.10 . . . . . . . . . . . . . . . . . . . . . . 178 4.A.5 Proof of Lemmas 4.3.2 and 4.3.7 . . . . . . . . . . . . . . . . 180 4.B Proof of Theorem 4.1.4 in the case σ X = 0 and σ C > 0 . . .

  using experimental data of the activity of the giant squid axon. It describes the ion exchanges K + , Na + and Cl through the membrane and their eects on the potential. A simpli- cation of this model is the FitzHugh-Nagumo model, which reduces the dimension: from four-dimensional model (for one neuron) with Hodgkin-Huxley equations, we obtain a two-dimensional model. It's a compromise between the biological accuracy and the mathematical simplicity.

  1.1) and (4.1.2) have well-dened solutions: Proposition 4.1.2. Let K X and K C satisfy Assumptions 4.1.1, with L X,max = ∞ and L C,max = ∞.

  Theorem 4.1.3. [Non uniform in time propagation of chaos] Let K X and K C satisfy Assumptions 4.1.1, with L X,max = ∞ and L C,max = ∞. There exist explicit C 1 , C 2 > 0, such that for all probability measures µ 0 on R 2 with nite second moment,

( 4 .

 4 1.1) and (4.1.2), under Assumption 4.1.1, and we begin with (4.1.1).

( 4 . 1 . 5 )

 415 The proof relies on the Lyapunov function dened in the next Section, and is given in Appendix 4.A.2. Then, by denoting T ∞ the explosion time of a solution of system (4.1.1)

Proposition 4 . 1 . 6 .

 416 If E(| X0 | 2 ) < +∞ and E(| C0 | 2 ) < +∞, then there exists C 0,1and C 0,2 such that:

Lemma 4 . 3 . 5 .

 435 For all t > 0

( 4 . 3 . 10 )

 4310 Proof of Theorem 4.1.4. With these four Lemmas, we can calculate 1 N N i=1

∂

  c H = c + α and ∂ x H = γx + β, so L µ H(z) =∂ x H(z)(x -x 3 ) + ∂ x H(z)K X * µ(z) -c∂ c H(z) + ∂ c H(z)K C * µ(z) + β)(x -x 3 ) -c(c + α) + (γx + β)K X * µ(z) + (c + α)K C * µ(z)

4.A. 3 5 )⇐⇒ 1

 351 Proof of Lemma 4.2.9We now prove that there are constants c, and δ such that Since for all u ≥ 0, 0< φ (u) ≤ 1, we have 0 < Φ (s) = s 0 φ (u) du ≤ s, i.e s/Φ (s) ≥ 1 . Therefore inf r∈]0,R] rφ (r) Φ (r) ≥ inf r∈]0,R] φ (r) = φ (R) .It is thus sucient for (4.A.The various conditions involving c invite us to consider 2 B = ηc. ≤ λ η 15λ + 32ηc(since c ≥ 0)⇐⇒ c ≤ λ32η -15 η .

Chapter 4 .

 4 Propagation of chaos for FHN neuronsWe choose to writeδ = (1 + δ) 1 + L X,max 1 -L C,max > 1 + L X 1 -L C Let usassume, for simplicity, that ≤ 1. It is sucient for this later condition to have c ≤ 2 B

4.A. 4

 4 Proof of Lemma 4.2.10 Let z, z ∈ R 2 . Proof of control of the L 1 distance : We have z -z 1 = |x -x | + |c -c | ≤ 1 min (δ, 1) |x -x | + δ|c -c | = 1 min (δ, 1)r(z, z ).

2 ≥

 2 δ 2 ) min (γ, 1) , 1 f (r(z, z )) 1 + H(z) + H(z ) .Proof of control of the L2 distance : We haver(z, z ) 2 = |x -x | + δ|c -c | 2 ≥ |x -x | 2 + δ 2 |c -c | min 1, δ 2 |x -x | 2 + |c -c | 2 . If r(z, z ) ≥ 1,we have, using (4.A.1) r(z, z ) 2 ≤ 16 (1 + δ 2 ) min (γ, 1) H(z) + H(z )

, 1 f

 1 (r(z, z )) 1 + H(z) + H(z ) .

  X est la diérence de potentiel, C est la capacité, I input est une intensité en entrée du système (avec des électrodes), et I model est l'intensité dans le neurone décrite par le modèle. Plan de la thèse et chaque chapitre 5 atteint une certaine valeur seuil X th . Lorsque X(t) atteint cette valeur, on considère que le neurone a une décharge et que X(t) revient à 0. Des modications de ce modèle ont été proposées, dès 1907, notamment le modèle Intégration et Tire avec fuite (Leaky integrate-and-re) qui prend en compte une fuite (ou diusion) des ions à travers la membrane du neurone. Dans ce cas, I model (t) est proportionnel à X(t).

			l'équation
	C	dX(t) dt	= I

input (t) -I model (t), où Le premier d'entre eux, attribué à Lapique en 1907 (bien qu'il ait en fait été nommé et décrit ainsi à partir des années 1960 d'après [17]), est le modèle Intégration-Et-Tire (Integrate-and-re). Dans ce modèle, I model (t) = 0 tant que X(t) n'a pas 1.2.

  De façon évidente, Chapitre 1. Introduction nous supposons E(τ ) > 0. loi des grands nombres (en supposant E[|W |] et E[τ ] nis)

	La Z t t	p.s. -→ t→∞	E[W ] E[τ ]	si et seulement si E	max S 0 ≤t<S 1	|r

t | < ∞ , et le théorème central limite (en supposant Var(W ) < ∞ et Var(τ ) < ∞)

  inf Plan de la thèse et chaque chapitre 9 dans (1.2.3). Quand η t est la distribution d'une variable aléatoire Y t (par exemple Z t /t) nous dirons que la famille (Y t ) t satisfait un Principe de Grandes Déviations.

			t→+∞	1 t	ln η t (O) pour tout ouvert O,	(1.2.2)
	et	-inf x∈C	J(x) ≥ lim sup t→+∞	1 t	ln η t (C) pour tout fermé C.

(1.2.3) Nous pourrons dire que (η t ) t∈R + satisfait un Principe de Grandes Déviations complet si (1.2.2) et (1.2.3) sont satisfaits, tandis que nous parlerons de Principe de Grandes Déviations faible quand les fermés C sont remplacés par des compacts C 1.2.

  Dénition 1.2.3. Supposons (X , d) est un espace métrique. Une famille de variables aléatoires Y n,t (n ∈ N) est une approximation exponentiellement bonne de Y t (toutes ces variables sont supposées dénies sur le même espace de probabilité (Ω, P)), si pour tout δ > 0 on a . Si X = R k équipé d'une norme quelconque, alors la même conclusion reste vraie lorsque Y n,t satisfait uniquement un Principe de Grandes Déviations faible. 3. Si J (dénie ci-dessus) est une bonne fonction de taux, telle que pour tout fermé F ,

	lim n→∞	lim sup
		inf

t→∞ 1 t ln P(d(Y n,t , Y t ) > δ) = -∞ . Le résultat clé pour notre étude est Théorème 1.2.4. Dans le cadre de la dénition 1.2.3, on suppose que Y n,t est une approximation exponentiellement bonne de Y t . Alors, les propositions suivantes sont vraies 1. Si Y n,t satisfait un Principe de Grandes Déviations complet, avec pour fonction de taux J n , alors Y t satisfait un Principe de Grandes Déviations faible avec la fonction de taux J(x) = sup δ>0 lim inf n→∞ inf d(y,x)<δ J n (y) . 2y∈F J(y) ≤ lim sup n→∞ inf y∈F J n (y)

10

  Chapitre 1. Introduction Déviations complet. Cependant, dans certains cas, l'étude de la fonction de taux J est dicile. Le lemme ci-dessous propose une alternative, en utilisant la notion de tension exponentielle, qui est facile à obtenir avec nos hypothèses. Lemme 1.2.5. Si Y t satisfait un Principe de Grandes Déviations faible avec une fonction de taux I et est exponentiellement tendue, i.e. pour tout α > 0, il existe un compact K α tel que alors Y t satisfait un Principe de Grandes Déviations complet et I est une bonne fonction de taux. Ce Lemme est une conséquence du Lemme 1.2.18 dans [25]. Le Théorème 1.2.4 et ce dernier Lemme 1.2.5 seront utilisés à la n de la preuve, pour prouver le Théorème principal 1.2.7.

	lim sup t→+∞	1 t	log P (Y t / ∈ K c α ) < -α,

  .2.7) On peut ensuite établir Théorème 1.2.7. Supposons que les Hypothèses 1.2.6 soient satisfaites. Soit J dénie par (1.2.6) et J par (1.2.7). On écrit m = E(W )/E(τ ). Si β 0 = +∞ (en particulier, si W est borné) alors Z t /t satisfait un Principe de Grandes Déviations complet avec J pour bonne fonction de taux. Nous avons également les inégalités suivantes Plan de la thèse et chaque chapitre 11 Si β 0 < +∞, nous avons pour tout a > 0 et tout κ ∈ (0, 1) Remarque 1.2.8. Les inégalités précédentes (1.2.8) et (1.2.9) restent en fait vraies avec la fonction J puisque J ≤ J. Néanmoins, comme J est beaucoup plus facile à obtenir, nous préférons écrire cette version.Remarque 1.2.9. En fait, nous considérons ici que r t = 0 dans la dénition de Z t .Cette hypothèse peut être relâchée si r t /t tend vers 0 assez vite, comme cela sera le cas pour les processus de Hawkes. Par exemple, si pour tout δ > 0

	et	lim sup t→+∞ lim sup t→+∞ 1 t ln P 1 t ln P Z lim sup 1 t ln P Z t t 1 t ln P t→+∞ Z t t ≥ m + a ≤ -min ≥ m + a ≤ -inf z≥m+a inf z≥m+κa J(z) , (1 -κ)β 0 a/2 , J(z), (1.2.8) (1.2.10) lim sup t→+∞ Z t t ≤ m -a ≤ -min inf z≥m-κa J(z) , (1 -κ)β 0 a/2 . (1.2.11) lim sup t→∞ 1 t ln P |r t | t > δ = -∞,

t t ≤ m -a ≤ -inf z≤m-a J(z).

(1.2.9) 1.2. alors Z t /t et (Z t -r t )/t sont dits exponentiellement équivalents et satisfont le même principe de grande déviation (si l'un des deux en admet un, l'autre en admet également un) et ont la même fonction de taux. C'est cette situation qui s'applique pour les processus de Hawkes, sous réserve d'hypothèse supplémentaire.

  En particulier, puisque des résultats sont déjà connus lorsque h est positive, nous souhaitons quantier la perte de points dûe à l'inhibition.

	La Proposition 1.2.13 nous permet d'établir une majoration de N h t par N h + t .
	Nous souhaitions également obtenir une minoration, ce que nous permet la Propo-
	sition suivante
	Proposition 1.2.15 (Minoration d'un processus de Hawkes).

∀t ≥ 0 Soit h une fonction satisfaisant les Hypothèses 1.2.14. Soit λ > 0, et dénissons g = -λ1 [0,L(h)] . On peut trouver un couplage de deux processus de Hawkes N h et N g , associés respectivement aux fonctions de reproduction h et g, et d'intensité de base λ, tel que pour tout t ≥ 0 : N h t ≥ N g t p.s. Il faut noter que ce résultat de comparaison est plus faible que la majoration via h + . En eet, nous n'avons pas N h ([s, t]) ≥ N g ([s, t]) pour tout s, mais seulement pour s = 0. Ce résultat motive l'étude détaillée de N g t pour g = -λ1 [0,A] , avec A > 0, qu'on appelle le cas d'annulation de l'intensité. Cette étude a été faite en particulier dans le Chapitre 3, dans la remarque 3.2.8, et permet d'obtenir des bornes pour certaines propositions.

  Cela signie que la durée entre le début d'une fenêtre et le premier saut de cette fenêtre suit une loi exponentielle.

	On peut écrire

La Proposition suivante nous permet d'apporter des propriétés importantes sur la loi jointe des (τ i , W i ) i . Il est néanmoins dicile d'apporter des informations plus explicites sans se restreindre à des cas spéciques. Proposition 1.2.16. Sous les Hypothèses 1.2.14, on a : i) les (τ i , W i ) i sont des variables aléatoires i.i.d., ii) pour i ∈ N * , les durées (U i 1 -S i-1 ) sont des variables aléatoires i.i.d. de distribution exponentielle E(λ).

  Théorème 1.2.20 (Théorème Central Limite). Soit h une fonction signée qui satisfait les Hypothèses 1.2.14 et N h le processsus de Hawkes donné par (1.2.13).

  On obtient nalement des résultats de grandes déviations, grâce au travail effectué sur les processus cumulatifs dans le chapitre précédent. Nous rappelons et adaptons les notations dans ce contexte.

Dénition 1.2.21. Nous dénissons la transformée de Cramer pour (a, b) ∈ R 2 , Λ * (a, b) = sup x,y ax + by -ln E e xτ 1 +yW 1 .

  Une fois cette étude sur τ et W menée, nous avons pu appliquer les théorèmes bien connus pour les processus cumulatifs à une approximation N h t du processus de Hawkes. Il a ensuite fallu être vigilants sur la loi de la quantité r t = N h

h + , et qui majorent respectivement τ et W . En ce qui concerne les moments exponentiels de τ + , nous avons pu comparer cette variable aléatoire à une quantité connue dans la théorie des les d'attente : il s'agit de la durée d'occupation de la le d'attente. Cette comparaison nous a permis d'utiliser des propriétés connues sur sa loi pour obtenir cette proposition. Les moments de W + ont été obtenus grâce à la forme W + = N h + ([0, τ + ]), et à l'utilisation des propriétés obtenues dans [8] dans le cadre des Hawkes linéaires. t -N h t . C'est cette quantité qui nous fait légèrement perdre sur la majoration dans les inégalités de grandes déviations (au lieu d'avoir une majoration par -(1 -κ)θ 0 a, on a une majoration en -(1 -κ)θ 0 a/2 dans (1.2.21) et (1.2.22)).

  'originalité de cette étude réside dans le lien entre processus cumulatif et processus de Hawkes. Néanmoins, ce lien repose sur les hypothèses que nous imposons aux processus de Hawkes, à savoir avoir une fonction de reproduction h à support com-

	1.2.1.4 Perspectives
	Finalement, ces deux chapitres exhibent une vision nouvelle des processus de
	Hawkes, sous les hypothèses étudiées, comme processus cumulatifs. Ce point de vue
	permet non seulement d'établir une loi des grands nombres et un théorème central
	limite pour les processus de Hawkes, mais également un principe de grandes dévia-
	tions. Nous démontrons ce principe pour les processus cumulatifs en démontrant un
	principe sur les mesures empiriques, suivant les travaux de [52].

L

pact. Si cette hypothèse ne pose en général pas de problème pour la modélisation en première approche (considérer qu'un processus n'a qu'une dépendance limitée dans le temps, en acceptant des temps très longs, n'est pas très restrictif ), il pourrait être intéressant d'étudier d'autres hypothèses moins fortes. Ainsi, Carl Graham

[START_REF] Graham | Regenerative Properties of the Linear Hawkes Process with Unbounded Memory[END_REF]

a

Il existe deux constantes explicites C 1 et C 2 strictement positives telles que pour toute mesure de probabilité µ 0 sur R 2 qui admet un second moment ni,

  1 et L 2 usuelles. Tout d'abord, il est nécessaire de vérier que les deux systèmes (1.2.23) et (1.2.24) ont des solutions bien dénies : Proposition 1.2.28 (Existence de solutions). Soit K X et K C satisfaisant l'Hypothèse 1.2.27, avec L X,max = ∞ et L C,max = ∞.Propagation du chaos non-uniforme en temps). Soit K X et K C deux fonctions satisfaisant les Hypothèses 1.2.27, avec L X,max = ∞ et L C,max = ∞. ⊗N , et où μt est une solution de (1.2.24) avec pour distribution initiale µ 0 . Propagation du chaos uniforme en temps). Soit C0 > 0 et ã > 0. Il existe un C K X ,K C > 0 explicite tel que, pour tout K X et K C satisfaisant les Hypothèses 1.2.27 avec L X , L C < C K X ,K C , il existe deux constantes B 1 et B 2 positives et explicites telles que pour toute mesure de probabilité µ 0

			Chapitre 1. Introduction
	Théorème 1.2.30 (
	Il existe une unique solution forte pour le système (1.2.23) et une unique solution forte pour le système (1.2.24).
	Nous pouvons maintenant énoncer le premier résultat de propagation du chaos :
	Théorème 1.2.29 (W 1 µ k,N t	, μ⊗k t	≤ C 1 e C 2 t k √ N	,
	pour tout k ∈ N, où µ k,N t neurones (Z 1,N t , . . . , Z k,N t avec pour distribution initiale (µ 0 ) Ce résultat, bien qu'à première vue plus faible que le résultat suivant, n'a pas est la distribution marginale, à l'instant t, des k premiers ) d'un système de N neurones suivant le système (1.2.23),
	de condition sur les coecients de Lipschitz des fonctions d'interaction. De plus, sa
	preuve permet de comprendre la démarche pour le théorème principal.
	Notre principal résultat est d'avoir réussi à enlever la dépendance en temps dans
	la borne supérieure précédente. Néanmoins, cette uniformité en temps s'accompagne
	d'hypothèses plus fortes sur les noyaux d'interaction.

  1.2.25) où || • || p décrit la distance usuelle L p sur R d . Cette distance étant dénie comme le minimum de l'espérance de la distance entre X et Y sur tous les couplages de (X, Y ) où X suit la loi µ et Y suit la loi ν, il sut de construire un couplage telle que cette quantité tend vers 0. Ainsi, nous construisons simultanément deux solutions de

	1.2. Plan de la thèse et chaque chapitre	25
	(1.2.23) et (1.2.24) qui se rapprocheront lorsque le nombre de neurones augmentera.
	Soit	Xi t , Ci t , pour i entre 1 et N , N copies indépendantes de solutions de
	(1.2.24) dirigées par des mouvements Browniens ( Bi,X t ) t 0 et ( Bi,C t ) t 0 indépen-
	dants. Un couplage de deux jeux de solutions alors directement d'un couplage des mouvements Browniens B et B. Xi t , Ci t i et X i,N t , C i,N t	i	découle
	Le premier choix, le plus naturel, a été popularisé par Sznitman [77]. Il s'agit
	du couplage synchrone qui consiste à choisir B = B. Ainsi, quand on considère
	l'évolution temporelle de Zi t -Z i,N t	= Xi t -X i,N t , Ci t -C i,N t	, le bruit s'annule.
	La contraction de la distance entre les processus repose alors uniquement sur la dé-
	rive (ou le drift) déterministe. Même s'il est, en général, alors nécessaire d'avoir une
	dérive fortement convexe, dans notre cadre, cette approche sut pour démontrer
	une propagation du chaos non uniforme en temps. On peut également voir qu'il y a
	une contraction déterministe susante lorsque Xi t -X i,N t	= 0. On va alors utiliser
	un couplage synchrone dans le voisinage de ce sous-espace.
	En dehors de ce sous-espace, nous utilisons le bruit pour rapprocher les proces-
	sus. On considère B = -B, qui maximise la variance du bruit, dans la direction
	orthogonale à ce sous-espace. Il s'agit alors d'un couplage par réexion (également
	nommé antithétique). Néanmoins, comme le bruit est symétrique, si nous considé-
	rons une distance classique de type r i t = | Xi t -X i,N t | + δ| Ci t -C i,N t | avec δ > 0, il
	n'y a pas de raison pour que ce couplage favorise une décroissance de r i t . C'est pour-
	quoi nous regardons f (r i t ), avec f une fonction concave spécique. Une diminution
	aléatoire a alors plus d'eet qu'une augmentation aléatoire de même valeur.

  Some additional technical topological lemmata. . . . . . . . 58 2.7 Proof of Theorem 2.4.3. . . . . . . . . . . . . . . . . . . . . . 62 2.8 Proof of Theorem 2.4.4. . . . . . . . . . . . . . . . . . . . . . 65 2.8.1 The contraction principle. . . . . . . . . . . . . . . . . . . . 65 2.8.2 Study of the rate function. . . . . . . . . . . . . . . . . . . . 66 2.8.3 End of the proof. . . . . . . . . . . . . . . . . . . . . . . . . 69 2.9 Proof of Theorem 2.2.5. . . . . . . . . . . . . . . . . . . . . . 70 2.9.1 Case A: β 0 = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.9.2 Case B: β 0 < ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.A Lemmata and Propositions from [52] . . . . . . . . . . . . . . 79 2.A.1 Complement to Section 2.5 The functional I n . . . . . . . . . 79 2.A.2 Complement to Section 2.6 Some additional technical topological lemmata. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.A.3 Complement to Section 2.7 Proof of Theorem 2.4.3 . . . . . . 88 2.A.4 Complement to Section 2.8 Proof of Theorem 2.4.4 . . . . . . 97 2.B Complementary proofs . . . . . . . . . . . . . . . . . . . . . . 99 2.B.1 Relative entropy . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.B.2 Lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . 100 2.1 Introduction.

	Chapter 2. LDP for cumulative processes
	2.6
	traitent à la
	fois le modèle FitzHugh-Nagumo et le modèle Hodgkin-Huxley.

  t and (Z t -r t )/t are exponentially equivalent and satisfy the same LDP (if (Z t -r t )/t have one) and have the same rate function.

  inf When η t is the distribution of some random variable Y t (for instance Z t /t) we shall say that the family (Y t ) t satises a LDP.Since J is lower semi-continuous the level sets {X, J(x) ≤ a} are closed. If in addition they are compact, then J is said to be a good rate function.

	t→+∞	1 t	ln η t (O) for all open subset O,	(2.2.2)
	and We shall sometimes say that (η t ) t∈R + satises the full LDP when (2.2.2) and (2.2.3) -inf x∈C J(x) ≥ lim sup 1 ln η t (C) for all closed subset C. (2.2.3) t t→+∞ are satised, while we will use weak LDP when C closed is replaced by C compact in
	(2.2.3).

  Y t satises a full LDP with rate function J.

			Chapter 2. LDP for cumulative processes
	The key result is then
	Theorem 2.2.2. In the framework of denition 2.2.1, assume that Y n,t is an expo-
	nentially good approximation of Y t . Then the following statements hold true.
	1. If Y n,t satises a full LDP with rate function J n then Y t satises a weak LDP
	with rate function
		J(x) = sup δ>0	lim inf n→∞	d(y,x)<δ inf	J n (y) .
	2. If X = R k equipped with any norm, then the same conclusion is true when
	Y n,t satises only a weak LDP.
	3. If J (dened above) is a good rate function such that for any closed set F ,
		inf y∈F	J(y) ≤ lim sup n→∞	inf y∈F	J n (y)
	The rst and last points in the previous Theorem are contained in [52] Theorem
	4.2.16. The second one is a consequence of the fact that closed balls are compact
	sets in R k . Usually, the Theorem is sucient to prove a full LDP. Nevertheless, it
	some cases, the study of the rate function J is dicult. The lemma below gives an
	alternative, using exponential tightness which is easy to obtain with our assump-
	tions.
	Lemma 2.2.3. If Y t satises a weak LDP with a rate function I and is exponentially
	tight, i.e. for all α > 0, there exists a compact set K α such that
	lim sup t→+∞	1 t	ln P (Y t / ∈ K c α ) < -α,
	then Y t satises a full LDP and I is a good rate function.
	This Lemma is a consequence of the Lemma 1.2.18 in [52].
	Theorem 2.2.2 as well as Lemma 2.2.3 will be used at the end of the article, in the
	section 2.9 which proves the main Theorem 2.2.5.
	2.2.2 Main results.
	Assumption 2.2.4. We will make the following set of assumptions:
	Denition 2.2.1. Assume that (X , d) is a metric space. A family of random vari-
	ables Y n,t (n ∈ N) is an exponentially good approximation of Y t (all these variables
	being dened on the same probability space (Ω, P)), if for all δ > 0 it holds
	lim n→∞	lim sup t→∞	1 t	ln P(d(Y n,t , Y t ) > δ) = -∞ .

then

  ln E(e λτ τ )

	Chapter 2. LDP for cumulative processes
	yielding the desired result.
	Study of P ct j=1 (W j -n) + > δt 2 . Denote as usual by ct the integer part
	of ct. We have
	≥ x + λ τ

  Remark 2.3.3. If W isn't bounded nor discrete, a double reduction can be done in one step. We obtain the same results as doing successively both the reductions, but it allows formulating the rate function more easily, when there is one.

	Let's consider

so that, since J τ grows to innity lim n→∞ lim sup t→∞ 1 t ln P Zn t t -Z t t > δ = -∞ . We have thus shown Lemma 2.3.2. If Assumption 2.2.4 is fullled, and W is almost surely bounded, Zn t /t dened above is an exponentially good approximation of Z t /t.

  By the proofs of Lemmas 2.3.1 and 2.3.2, we nally obtain Lemma 2.3.4. If Assumption 2.2.4 is fullled, for all δ > 0,

	lim n→∞	lim sup t→∞	1 t	ln P	Z t t	-

  Therefore µ ∈ ∆n . Chapter 2. LDP for cumulative processes Remark 2.6.3. Here again one can replace {1, ..., n} by a compact subset W. ♦

	We nally set two important results. The rst one is identical to Lemma 2.6
	and Lemma 2.7 in [52]:

1

-α 1-αβ η, ᾱ = αβ and π(dτ, dW ) = π(dτ, dW |τ < ∞).

  2.7. Proof of Theorem 2.4.3. 63 whose proof is unchanged in our case (see Lemma 2.A.8). Since {ν ∈ M 1 ((0, +∞] 2 × {1, ..., n}) , ν(1/(a + b)) ≤ M } is compact (see lemma 2.4.1), exponential tightness follows.

  ) . according to[START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] Lemma 5.4, M ε t is an exponentially good approximation of M t (see Lemma 2.A.13). It follows that µ ε,n t (ϕ ε ) is an exponentially good approximation of Z n

	But

t /t (see Lemma 2.A.14) so that, nally, thanks to Theorem 2.2.2, Z n t /t satises a weak LDP with rate function

  t is an exponentially good approximation of Z t /t (see Lemma 2.3.1, 2.3.2 or 2.3.4 depending on the approximation strategy). Combining the LDP principle obtained for Z n t /t (Theorem 2.4.4) with Theorem 2.2.2 (1) we obtain that Z t /t satises a weak LDP with rate function

  We handle the rst term with the full LDP for Z n t /t with the rate function J n (Theorem 2.4.4) and the second term with Lemma 2.3.1 (or 2.3.2, or 2.3.4 depending on the approximation strategy). We then have

												we obtain
	lim sup t→∞	1 t	ln P	Z t t	≥ m + a
		≤ max lim sup t→∞	1 t	ln P	Z n t t	≥ m + κa , lim sup t→∞	1 t	ln P	Z t t	-	Z n t t	≥ (1 -κ)a .
				lim sup t→∞	1 t	ln P	Z t

  O be an open subset of M 1 ((0, +∞] 2 × {1, ..., n}). We apply this last inequality for A = O ∩ ∆ n

	92			Chapter 2. LDP for cumulative processes
			M,g,δ :
	lim sup t→+∞	1 t	ln P n t (O) ≤ lim sup t→+∞	1 t	ln 2 max(P n t (O ∩ ∆ n M,g,δ ), P n t ((∆ n M,g,δ ) c ))
			≤ max lim sup t→+∞	1 t	ln(P n t

5. Thus we have: lim sup t→∞ 1 t ln P n t (A) ≤ -inf µ∈A µ(f ).

Let

  2.A.10 (Lower bound). For each open setO ⊂ M 1 ((0, +∞] 2 × {1, ..., n}),The proof of this Proposition will be given at the end of this part. We can now give the proof of the lower bound: Proof of the lower bound. Let µ ∈ ∆ n and V be an open neighborhood of µ in the weak topology. By the proposition 2.A.11, there exists Q t such that Q t δ µ and:

	lim inf	1 t	ln P n t (O) ≥ -inf
	lim sup

µ∈O

I n (µ).

The proof of this bound is much more simple than the upper bound. We only need the following Proposition: Proposition 2.A.11 (Equivalent of Proposition 4.2). For every µ ∈ ∆ n , there exists a family Q t of probability measures on M 1 ((0, +∞] 2 × {1, ..., n}) such that Q t δ µ and lim sup

t→+∞ 1 t H(Q t |P t ) ≤ I n (µ). t→+∞ 1 t H(Q t |P n t ) ≤ I n (µ).

  kε, and nally M t-kε ≤ m = M ε , for s ≤ t and k ∈ N, {M t -M s > k} = {S Ms+k ≤ t}. We dene ( Ŝk = S Ms+k -S Ms ) k≤1 . It has the same law as (S k ) k≤1 . Then{S Ms+k ≤ t} = {S Ms + Ŝk ≤ t} ⊂ { Ŝk ≤ t -s}.

			t . Eventually,
	since k ≤ Ct, M t-Ctε ≤ M ε t . Therefore
	P (M t -M ε t > tδ) ≤ P (M NowThen (Ct 2 ) 2
	P (M t -M ε t > tδ) ≤ P S C→∞ (Ct 2 ) 2 lim ε→0 lim sup t→+∞ 1 t ln 2 max (Ct 2 ) 2 P S Finally
	lim ε→0	lim sup
	Lemma 2.A.14. µ ε,n t (ϕ ε ) is an exponentially good approximation of Z n t /t, i.e. for
	all δ > 0 it holds
		lim ε→0	lim sup

t -M t-Ctε > tδ) + P(S Ct ≤ t). tδ ≤ Ctε + P(S Ct ≤ t). Therefore lim ε→0 lim sup t→+∞ 1 t ln P (|M t -M ε t | > tδ) ≤ lim sup tδ ≤ Ctε , P(S Ct ≤ t) ≤ max lim ε→0 lim sup t→+∞ 1 t ln P S tδ ≤ tε , lim sup C→∞ lim sup t→+∞ 1 t ln P(S Ct ≤ t) . By the Markov inequality, we deduce P S tδ ≤ tε = P e -S tδ /ε ≤ e -t ≤ e t+ tδ ln E(e -τ 1 /ε ) . t→+∞ 1 t ln P S tδ ≤ tε ≤ lim ε→0 lim sup t→+∞ 1 t t + tδ ln E(e -τ 1 /ε ) = -∞ and lim sup C→∞ lim sup t→+∞ 1 t ln P(S Ct ≤ t) ≤ lim sup C→∞ lim sup t→+∞ 1 t t + tC ln E(e -τ 1 = -∞. t→∞ 1 t ln P µ ε,n t (ϕ ε ) -

  there is either at least one other jump of N h in (U g j , U g j+1 ) or no other jump. If there is no other jump, then

  The rst result deals with this problem.

Proposition 3.2.6. Let h be a signed function satisfying Assumptions 3.2.3. Let us consider the Hawkes process N h and the i.i.d. couples of random variables (τ i , W i ) dened in (3.2.4)-(3.2.5).

  Remark 3.2.14. Once again we may get an explicit expression for the rate function in the canceling intensity case h = -λ 1 [0,A] . Since W 1 = 1 and τ 1 -A is an exponential variable with parameter λ, we have

										.2.10)
	Similarly
	lim sup t→∞	1 t	ln P	N h t t	≤ m -a ≤ -min	inf m-z≤κa	J(z) ,	(1 -κ) 2	θ 0 a .
										(3.2.11)
	for κ ∈ (0, 1).
	βΛ * 1 β	,	m β	= sup x,y

  In blue, the intensity function t → Λ h (t); in red, the jumps times. The axis below indicates the Dirac measures of the process.We can summarize the results of this two cases and apply Theorem 3.2.9 to

	3.3. One more example with explicit calculations: canceling intensity with delay. 117
		Λ h (t)			Λ h (t)
	λ	A			λ	r
					t	t
					N h	N h
	(a) Example of Hawkes process : can-	(b) Example of Hawkes process : can-
	celing intensity without delay, h =	celing intensity with a delay, h =
	-λ 1 [0,A] .			-λ 1 [r,r+A] .
	Figure 3.3 Comparison of Hawkes processes with or without a delay in the canceling
	of the intensity:
		lim t→∞	N h t t	=	λ(1 + λr) λA + 2λr + e -r a.s.

obtain Proposition 3.3.1. Let us consider A > 0 and r ≥ 0. The Hawkes process associated with h = -λ 1 [r,r+A] satises

  t 2 , ... an increasing sequence of times such that t i -→

	i→+∞ > t i Chapter 3. Limit theorems for Hawkes processes +∞. For a xed i, we have for t large enough P M h t > t 2 = P M h t t > t ≤ P M h t t Since lim sup t→+∞ 1 t ln P M h t t It follows, lim sup t→+∞ 1 t ln P M h t t > t = -∞. Eventually, lim sup t→+∞ 1 t ln P W M h t +1 t > δ ≤ lim sup t→+∞ 1 t ln P M h t > t 2 + (t 2 + 1)P (W 1 > δt) ≤ lim sup t→+∞ ln 2 t + max 1 t ln P M h t t > t , 1 t ln (t 2 + 1)P (W 1 > δt) > t 126 ≤ max lim sup t→+∞ 1 t ln P M h t t > t , lim sup t→+∞ ln(t 2 + 1) t + 1 t ln P (W 1 > δt)

i ≤ -J τ 1 (t i ) ≤ -t i -η 0 .

  2 t . Quick result : non uniform in time propagation of chaos We start by proving Theorem 4.1.3, a non uniform in time propagation of chaos, as it highlights the basic strategy behind a coupling argument. Some of the following expressions will be used in the proof of Theorem 4.1.4, in Section 4.3

		(4.1.6)
	The proof is very similar with Proposition 4.1.5 and is in Appendix 4.A.2.
	4.1.4 We consider a synchronous coupling between (Z i,N t ) i and ( Zi t ) i , i.e. Bi,X t and Bi,C t = B i,C t . We have	= B i,X t

  as given in Assumption 4.1.1. In the case of uniform in time propagation of chaos, the inequalities L X and L C must satisfy are listed B, H, H, α X , α C , β X , β C : H (resp. H) is a Lyapunov functions given in (4.2.2) (resp. (4.2.9)). Its main property involves parameters λ and B (resp. λ and B), as can for instance be seen in (4.2.4) (resp. (4.2.14)). α X , α C , β X and β C are intermediate constants given in Lemma 4.2.3, (see (4.2.22)) is a concave function, the denition of which involves g, φ, Φ (see Assumption 4.2.8 for these last three). Function G (see (4.2.27)) is then used to dene ρ (see (4.2.26)), the semimetrics we consider in the end. All those notations thus refer to the modied distance we consider. These functions will be applied to r a modication of the usual L 1 distance (see equation (4.2.25)). Then, parameters δ, R, , C C 1 , C 2 , C z : constants used to quantify the control our modify distance has over the usual L 1 and L 2 distance (see Lemma 4.2.10 for the control and Assumption 4.2.8 for explicit values), φ rc , φ sc , ξ :

	in Assumption 4.2.8,
	W p :	the usual Wasserstein distance associated to the L p distance (see
	(4.1.3)),
	a, ã, C 0 :	constants used to give an exponential initial moment to the
	problem (see the assumptions of Theorem 4.1.4 and Section 4.2.3),
	λ, B, c :	a contraction rate (see Assumption 4.2.8),
	r, f, g, φ, Φ, G, ρ, δ, R, , C f (f ) (f ) 1 , C (f ) 2 : 1 , and C (f ) 2 are used to dene such functions (see Assumption 4.2.8
	for some explicit values),
	R 0 , φ min :	intermediate constants (see Assumption 4.2.8),
	C 0 :	uniform in time bound on the second moment of the processes (see
	Lemma 4.2.5),

  For all δ > 0 there is C r,H > 0 such that for all x, x , c, c ∈ R, we have|x -x | + δ|c -c | 2 ≤ C r,H (H(x, c) + H(x , c )), (iv) A direct consequence of the previous point is that for all B ∈ R, λ > 0 and δ > 0, there is R ≥ 0 such that, for x, x , c, c ∈ R satisfying |x -x | + δ|cc | ≥ R, we have H(z) + H(z ) ≥ 64B15λ . An explicit value of R is given by R = 1024(1+δ 2 )B 15λ min(γ,1) . The rst two points are consequences of direct calculations. The last two points are proved in Appendix 4.A.1. Lemma 4.2.3 (Lyapunov's property of H). H, dened in (4.2.2), is a Lyapunov function. In fact, let λ ∈ R such that

						2 +	H 0 2	.
	Lemma 4.2.2. We have
	(i) For all x, c ∈ R, we have H(x, c) ≥ γ 4 x 2 + c 2 4 ≥ 0 and H(x, c) ≤ γx 2 +c 2 + 3 2 H 0 ,
	(ii) For all x, c ∈ R, we have H(x, c) ≥	1 2 max(γ,1) (γx + β) 2 + (c + α) 2 ,
	(iii) L X 8	+ L C 2 +	3 32	< 1 -	λ 2	,	(4.2.3)
	then there exists B > 0 such that for all (x, c) ∈ R 2 , for all probability distribution
	µ on R 2 ,

  We obtain the corresponding Proposition 4.1.6 for X and Cwith Inequation (4.2.6). The proof of these propositions is given in Appendix 4.A.2.It also yields the following uniform in time bound on the second moments when there exists λ > 0 satisfying (4.2.3).

			.2.8)
	We obtain Proposition 4.1.5 on the second moments of X i,N t	and C i,N t	thanks to
	the Inequation (4.2.7). Lemma 4.2.5. Provided the interaction kernels satisfy

  .2.14) and (4.2.20) below.

	Lemma 4.2.6. We have, for all z ∈ R 2
		H(z) exp a H(z) ≥ H(z) ≥ exp a H(z) -	2 a 2 exp	a 2 2	-1 ,
								(4.2.10)
	2 a	H(z) exp a H(z) ≥ H(z) ≥	1 a	H(z) exp a H(z) -	1 a 2 (e -2) ,
								(4.2.11)
						H(z) ≥H(z)	(4.2.12)
	We may calculate, using Lemma 4.2.2 and Equation (4.2.4)
		Let a > 0, such that a ≤ ã/ 2	√	2 max	√ γ, 1 . This choice of a is only neces-
	sary for further Propositions and Lemmas, in Section 4.3.
		Let us consider for all z ∈ R 2 ,
		H(z) =	0	H(z)	exp a √	u du =	2 a 2 exp a H(z) a H(z) -1 +	2 a 2 . (4.2.9)
	Direct calculations yield the following technical lemma.

  Given any η > 15 and δ > 0, consider the following set of parameters

	4.2. Preliminaries	151
	Assumption 4.2.8.

  Propagation of chaos for FHN neuronsSince by Assumption 4.1.1, L X < L X,max and L C < L C,max , we know

	152	Chapter 4. L X 8 + L C 2 +	3 32	< 1 -	λ 2	.
	By (4.2.23) and by considering greater C z for instance, and using L X ≤ λ 128Cz
	and L C ≤ λ 128δCz , we have
		1	and	L C ≤ min	λ 128δC z	,	λa 512 δC z	,	c 2δC 1	.
										(4.2.23)

  Proof of Theorem 4.1.4 in the case σ X > 0

	161
	(4.2.19), to control the sum:

2,i

t with compensating terms, and in particular Lyapunov functions, which appears with the use of (4.2.14) and 4.3.

  4 , we have (y 1 + y 3 ) (y 3 e ay 3 + y 4 e ay 4 ) ≤ 2 y 2 1 e ay 1 + y 2 3 e ay 3 + y 2 3 e ay 3 + y 2 4 e ay 4 ,

	we obtain for this last sum

  .3. Proof of Theorem 4.1.4 in the case σ X > 0 γδ + L X + δL C )ξ -(δ -δL C -1 -L X ) )ξ 1 + γδ + L X + δL C + 1 -L C -

													169
	Since r i t = |X i,N t (r i t -ξ)/δ. Since δ > -Xi t | + δ|C i,N t 1 + L X 1 -L C , we obtain -Ci t | and |X i,N t	-Xi t | < ξ, we have |C i,N t	-Ci t | ≥
	Ki t 1 Reg i 2	≤G i t 2cf (r i t ) + ϕ rc |X i,N t	-Xi t |	2	2σ 2 x f (r i t ) + C	(f ) 1 + C	(f ) 2	σ 2 x r i t f (r i t )
			+ f (r i t ) (1 + r i t -ξ δ
			+ f (r i t )4	B
			≤ϕ rc |X i,N t		-Xi t |	2	G i t 2σ 2 x f (r i t ) + C	(f ) 1 + C	(f ) 2	σ 2 x r i t f (r i t )
			+ G i t f (r i t 1 + L X δ
			+ G i t (2c + 4 B)f (r i t ) -r i t f (r i t ) 1 -L C -	1 + L X δ	.
	By (4.3.12),
		2σ 2 x f (r i t ) + C	(f ) 1 + C	(f ) 2	σ 2 x r i t f (r i t )
	and by Lemma 4.2.9
				≤ 0. 2c + 4 B ≤ 1 -L C -	1 + L X δ	min r∈]0,R]	f (r)r f (r)	,	4.3.12)
	we obtain
	Eventually, in this region of space
			Ki t 1 Reg i 2							1 t 1 Reg i Ki	≤ 0.
	4.3.4.2 Region 2 : |X i,N t	-Xi
	In this region, we can write Ki t as
	Ki t 1 Reg i 2	=G i t 2cf (r i t ) + ϕ rc |X i,N t	-Xi t |	2	2σ 2 x f (r i t ) + C	(f ) 1 + C	(f ) 2	σ 2 x r i t f (r i t )
			+ f (r i t ) (1 + γδ + L X + δL C )|X i,N t	-Xi t |
			-(δ -1 -L X -δL C )|C i,N t	-Ci t | -G i t f (r i t )|(X i,N t ) 3 -( Xi t ) 3 | + f (r i t )4	B
			-f (r i t )	  λ 16	H( Zi t ) +	λ 16	H(Z i,N t ) +	λ 16N	N j=1	H( Zj t ) +	λ 16N	N j=1	H(Z j,N t )	  .

t | < ξ and r i t ≤ R. 4= -(2c + 4 B)Φ(r i t ) -f (r i t )r i t (1 + δγ + L X + L C ) ≤ 0,

  Proof. We have H(z) ≥ γ4 x 2 + c 2 4 ≥ 1 4 min (γ, 1) x 2 + c 2 . Thus r(z, z ) 2 = |x -x | + δ|c -c | 2 ≤2|x -x | 2 + 2δ 2 |c -c | 2 ≤4x 2 + 4x 2 + 4δ 2 c 2 + 4δ 2 c 2 ≤4(1 + δ 2 )(x 2 + c 2 ) + 4(1 + δ 2 )(x 2 + c 2 ) Proof ofLyapunov's property of H and its consequences Lyapunov's property Proof of Lemma 4.2.3. First, we prove (4.2.4). It will yield (4.2.5) as we describe it at the end. We notice

			≤16	(1 + δ 2 ) min (γ, 1)	H(z) + H(z )
	4.A.2
	Ki t 1 Reg i 3	=f (r i t )	  2cG i t + 4 B -	λ 16	  H( Zi t ) + H(Z i,N t ) +	1 N	N j=1	H( Zj t )
				+	1 N	N j=1		H(Z j,N t )	    .

  γx + β)K X * µ(z) ≤L X (γ|x| + β)(|x| + |c| + E µ (|X|) + E µ (|C|)) ≤L X γ|x| 2 + γ|x||c| + γ|x|E µ (|X|) + γ|x|E µ (|C|) + β|x| + β|c| +βE µ (|X|) + βE µ (|C|)] ,and using Young's inequality ab ≤ α 2 a 2 + 1 2α b 2 (α = 16 when we separate x term and c term, and α = 1 otherwise on the various terms we get(γx + β)K X * µ(z) ≤L X γ|x| 2 + 8γ 2 |x| 2 +The idea is then to bound λH(z) + L µ H(z), by distinguishing 3 types of terms: we isolate terms in E µ (|C|) 2 -c 2 , E µ (|X|) 2 -x 2 , and we group polynomials terms. Then, we notice the polynomial is upper bounded by a constant A.≤ λH 0 + 17β 2 L X + 17α 2 L C -γx 4 -βx 3 + (1 + λ)βxAccording Assumption 4.1.1 and Remark 4.2.7 coecient of c 2 is negative, and there exists A ≥ 0 such that -γx 4 -βx 3 + (1 + λ)βx

	so
		(γx + β)K X * µ(z) ≤L X 17β 2 + |x| 2 1 2	+	3 2	γ + 16γ 2 +	|c| 2 16
																+E µ (|X|) 2 γ 2	+	1 2	+	E µ (|C|) 2 16	.
	Likewise
	(c + α)K C * µ(z) ≤L C 17α 2 +	17 2	|x| 2 + |c| 2 3 2	+	1 16	+ E µ (|X|) 2 17 2
												+E µ (|C|) 2 1 2	+	1 32	.
							σ 2 x γ 2	-	σ 2 c 2
	=λ c + 1 2 γx 2 + βx + 1 2 1 + λ 2 γ + L X	1 2	+	3 2	γ + 16γ 2 +	17 2	L C + L X	γ 2	+	1 2	+	17 2	L C x 2
	+	L X 16	+ L C		3 2	+	1 16			+	L X 16	+ L C	1 2	+	1 32	-1 -	λ 2	c 2 -(1 -λ)αc
	+	L X 16	+	L C 2	+		L C 32	E µ (|C|) 2 -c 2
	+	γ 2	L X +	1 2	L X +		17 2 L L X 8	+ L C 2 +	3 32	< 1 -	λ 2	,
																|c| 2 32	+	γ 2	|x| 2 +	γ 2	E µ (|X|) 2 + 8γ 2 |x| 2
	+ (1 +	λ 2	)γ +	L X 2	+ + E µ (|C|) 2 32 3L X 2 γ + 16γ 2 L X + + β 2 2 + |x| 2 2 17 + 8β 2 + 2 L C + γ 2	|c| 2 32 L X + +	β 2 2 1 2 L X + + 1 2	E µ (|X|) 2 17 2 L C x 2
	+	L X 16	+	3L C 2	+	+ 8β 2 + L C 16 + L X 16	E µ (|C|) 2 32 + L C 2 +	, L C 32	-1 -	λ 2	c

Thus λH(z) + L µ H(z) -2 + αc + H 0 + (γx + β)(x -x 3 ) -c(c + α) + (γx + β)K X * µ(z) + (c + α)K C * µ(z) C E µ (|X|) 2 -x 2 the 2 -(1 -λ)αc ≤ A.

  Since Lemma 4.2.2 (i), i.e. EH Z i,N β/λ < 0 with this choice of λ.Then, there exists C t < ∞, such that for each i ≤ NE |X i,N t | 2 + |C i,N t | 2 ≤ C t .Proof of Proposition 4.1.6. K X and K C are Lipschitz with constants L X and L C respectively. We do not assume any bound on these constants.In fact, when L X and L C are wisely bounded, the bound of the second moment can be uniform in time. It's the interest of the Lemma 4.2.5, which is proved below.Proof of Lemma 4.2.5. We assume E(| Xi

	with							t	≥ γ 4 E |X i,N t | 2 + 1 4 E |C i,N t | 2 and
	EH Z i,N 0 γ 4N N i=1	≤ γE |X i,N 0 | 2 + E |C i,N 0 | 2 + 3 2 H 0 , we obtain E |X i,N t | 2 + 1 4N C 0,1 = 4 max(γ, 1) γE | Xi 0 | 2 + E | Ci 0 | 2 + 3 2 H 0 -N i=1 E |C i,N t | 2 C 0,2 = -λ = 2 L X 8 + L C 2 + 3 32 .	B λ	,
				≤e -λt 1 N	N i=1	EH Z i,N 0	-	B λ	+	B λ
				≤	e -λt N	N i=1	γE |X i,N 0 | 2 + E |C i,N 0 | 2 +	3 2	H 0 -e -λt β λ	,
	The Inequation (4.2.7) gives us L X 8 + L C 2 +	3 32	< 1 -	λ 2	.
	d dt For instance, let λ = -2 L X 1 N N i=1 EH Z i,N t 8 + L C 2 + 3 ≤ B -λ 32 lemma, we obtain	1 N < 0. Then, using the Gronwall's N EH Z i,N t i=1
	for λ ∈ R which satises Thus, since β/λ < 0 with this choice of λ, we get L X 8 + L C 2 + 3 32 < 1 -EH Zi t -B λ ≤e -λt EH Zi 0 -λ . 2	B λ	.
	For instance, let λ = -2 L X 8 + L C 2 + 3 32 lemma, we obtain γ 4 E | Xi t | 2 + 1 4 E | Ci t | 2 ≤e -λt γE | Xi < 0. Then, using the Gronwall's 0 | 2 + E | Ci 0 | 2 + 3 2 H 0 -B λ .
	1 N	N i=1	EH Z i,N t	-	B λ	≤e -λt 1 N	N i=1	EH Z i,N 0	-	B λ	.

since We assume E(| Xi 0 | 2 ) < +∞ and E(| Ci 0 | 2 ) < +∞. The proof is really similar to the last proof. The Inequation (4.2.6) gives us d dt EH Zi t ≤B -λEH Zi t . for λ ∈ R which satises Finally E | Xi t | 2 + | Ci t | 2 ≤C 0,1 e C 0,2 t 0 | 2 ) < +∞ and E(| Ci 0 | 2 ) < +∞.

  and, if r(z, z ) ≥ 1, recall Lemma 4.2.2.Independence with respect to L X and L C The a priori bounds L X ∈ [0, L X,max ] and L C ∈ [0, L C,max ] allow us to bound φ(R) independently of L C and L X by φ min (and we also use g(R) ≥ 1 2 ), thus giving us constant C 1 , C 2 and C z independent of L C and L X . 4.A.5 Proof of Lemmas 4.3.2 and 4.3.7 Proof of Lemma 4.3.2. Let's prove there exists an uniform in time bound on E(G i t ) and E[(G i t ) 2 ]. First, let's recall the denition of G with Equation (4.2.27):The idea is to bound the dierent expectation in terms of the expectation at timet = 0. Since E(e ã(|X 0 |+|C 0 |) ) is nite, we know that for each k ∈ N, E(|X 0 | k ) and E(|C 0 | k ) are also nite. We deduce that for each k ∈ N, for each j ≤ N , E[H( Zj 0 ) k ] and E[H(Z j,N 0 ) k ] are nite.In fact, to bound uniformly in time the rst moment, we only have to bound E( H(Z j,N t )) and E( H( Zj t )) for each j ≤ N . Let's begin with Zj . By (4.2.16), we Now, it is enough to prove that there exist C such that for all z ∈ R 2 exp 2a H (z) ≤ C × e ã(|x|+|c|) . fact, from the denition of H in (4.2.2), we have C is a constant independent of z. Finally, since max (a

	4.A. Various technical lemmas					181
	and thus z -z 1 ≤ We already know E H Zj 1 φ(R)g(R) min 1, 0 is bounded. In 2 H(z) = 4 max 1 γ , 1 2 √ 2 γ x + β γ	f (r(z, z )) 1 + 2 + (c + α) 2 + H 0	H(z) +	H(z ) .
			≤ 2γ x +	β γ		+	√	2 |c + α| + H 0
			≤ 2γ|x| +	√	2|c| +	1 a	ln C,
	G i t = 1 + H( Zi t ) + H(Z i,N t ) +	N	N j=1	H(Z j,N t ) +	N	j=1 N	H( Zj √ 2γ, a	√	2) ≤ ã, we
	have
	d dt	E H Zj t	≤ B -		λ 4	E H Zj t	,
	then by Gronwall's lemma,
	E H Zj t	≤	4 λ B	+ e -λ 4 t E H Zj 0	-	B λ 4
			≤ max E H Zj 0	,	B λ 4	.
	By Inequation (4.2.10) in the Lemma 4.2.6, we deduce the following inequation and
	z -z 1 ≤ we apply Cauchy-Schwarz 4 γ ≤4 max H(z) + 1 E H Zj 0 ≤E H Zj 0 exp a H Zj 4 H(z ) + 4H(z) + 4H(z ) γ , 1 0 H(z) + H(z ) γ ≤ 4 max 1 γ , 1 f (r) φ(R)g(R) 1 + H(z) + ≤E H Zj 0 1/2 2 1/2 E exp 2a H Zj 0	H(z ) , (4.A.6)

t ).

where have exp 2a H (z) ≤ C × e ã(|x|+|c|) .

  , by the denition of H in (4.2.9) in Subsection 4.2.3, As for the rst moment, the study of Z j,N t is very similar to the one of Zj t . Here, we only focus on the second one. To bound E[ H( Zj t ) 2 ], we only have to bound E exp 2a H( Zj Zj t ) . First, let's notice for all z ∈ R 2 , 2H(z) ≤ H(2z) + H 0 . Then, by (4.2.10) from Lemma 4.2.6, we know∀z ∈ R 2 , exp 4a H(z) ≤ exp a H(2z) ≤ H(2z) + C, where C is a constant. Let's denote G(z) = H(2z). A study on the generator shows that ∀z ∈ R 2 , L µ G(z) ≤ 4L µ H(z). .A.Various technical lemmas 183 where we have used (4.2.10) in (4.2.14) for the last inequality. Since L µ H is a We can make the same decomposition by Cauchy-Schwarz than above in (4.A.6). Then, since H is a polynome, E H(2 Zj 0 ) 2 is also bounded uniformly in time. ) 2 ) is bounded uniformly in time. Finally, we deduce E (G i t ) 2 is bounded uniformly in time. Proof of Lemma 4.3.7. Using ∂ x H(z) = γx + β, we have∂ x H(Z i,N t ) -∂ x H( Zi Lemma 4.2.2 (ii), we have H(z) ≥ 1 2 min 1 γ , 1 (γx + β) 2 .By the mean value theorem, for all y 1 ≤ y 2 in R, there exists y 3 ∈ [y 1 , y 2 ] such that: e ay 1 -e ay 2 = a(y 1 -y 2 )e ay 3 . In particular, |e ay 1 -e ay 2 | ≤ a|y 1 -y 2 |(e ay 1 + e ay 2 ). Now, by Lemma 4.2.2 (i), we have H(z) ≥ γ 4 x 2 + 1 4 c 2 and since X i,N , we can nally use Lemma 4.2.6, and more precisely (4.2.10) and (4.2.11), we

	Then by the denition of H we get
	H(z) 2 = positive function, we have: 2 a 2 exp a H(z) a H(z) -1 + obtain ≤2 2 2 a 4 exp 2a H(z) a H(z) -1 2 + 2 2 a 2 E( G( Zj t )) ≤E( G( Zj 0 )) + 4 B ≤ E( H(2 Zj H(Z i,N t ) -H( Zi t ) = 1 2 γ (X i,N t ) 2 -( Xi t ) 2 + β(X i,N t -Xi 2 t ) |∂ x H(Z i,N t ) -∂ x H( Zi t )| 0 )) + 4 B. 2 2 a 4 ≤ 8 a 2 exp 2a H(z) 2a 2 H(z) + 2 + + 1 2 (C i,N t ) 2 -( Ci t ) 2 + α(C i,N t -Ci t ) ≤r i t γ + a 2 max (γ, 1) β + α δ H(Z i,N t ) + H( Zi t ) + 4 a 2 exp 8 a 2 . Since 2 max (a √ 2γ, a √ 2) ≤ ã, we have E exp 2a H 2 Zj 0 is bounded. Thus a 2 -1 2 we deduce E( H( Zj ≤ 1 2 γ X i,N t -Xi t X i,N t + Xi t + β X i,N t -Xi t + + ar i t 2 max (γ, 1) √ γ + 1 δ 2a H(Z i,N t ) + 2 a (e -2) + 2a H( Zi t ) + 2 (e -2) a 1 2 C i,N t -Ci t C i,N t + Ci t + α C i,N t -Ci t . ≤r i t H(Z i,N t ) + H( Zi t ) γ + a 2 max (γ, 1) β + α δ + 2a 2 2 max (γ, 1) √ γ +	1 δ
	and C i,N t + r i -Ci t γ + a 2 max (γ, 1) β + t ≤ r i t /δ, we get +4 2 max (γ, 1) √ γ + 1 δ (e -2) . α δ a 2 exp 4	a 2 2	-1	t	-Xi t ≤ r i t
	1/2 t ) + H( Zi H(Z i,N t ) + β (given in Assumption 4.2.8) the following constants + Xi t + β ≤ r i t √ γ and C γ X i,N t We denote by C X i,N t -Xi t 1 2 (f ) 1 (f ) 2
	and	C i,N t C (f ) -Ci t +4 2 max (γ, 1) 1 2 C i,N t + Ci t + α ≤ √ γ + 1 δ	α δ (e -2) r i t δ	4 a 2 exp H(Z i,N	a 2 2	-1
		C	(f ) 2 = 4 γ + a 2 max (γ, 1) β +	α δ	+ 2a 2 2 max (γ, 1)	√ γ +	1 δ
	Since X i,N t By the denition of G i -Xi t ≤ r i t , t and since G i α δ r i t + t ≥ 1, we obtain √ γ + 1 δ r i t	H(Z i,N t ) + H( Zi
	Finally,	γX i,N t	-γ Xi |∂ x H(Z i,N t ) -∂ x H( Zi t )| ≤ r i t	G i t C	(f ) 2 4	+ r i t G i t	C	(f ) 1 4	,
	|∂ x H(Z i,N t ) -∂ x H( Zi t )| and eventually
	Thus, by Dynkin's formula, E( G( Zj t )) ≤E( G( Zj 0 )) + E 2H( Zi t ) min 1 γ , 1 H(Z i,N t ) -H( Zi t 0 L µ G( Zj s )ds t ) exp a H(Z i,N t ) + a 2 max (γ, 1) √ γ + 1 δ r i t H(Z i,N t ) + H( Zi t ) exp a H(Z i,N t ) + exp a H( Zi t ) ≤r i t γ + a 2 max (γ, 1) β + α δ exp a H(Z i,N t ) + exp a H( Zi t ) By ≤a ≤γr i t exp a H(Z i,N t ) + exp a H( Zi t ) + a 2 max (γ, 1) β + α δ r i 2 1 + 1 σ 2 x ϕ rc |X i,N t -Xi t | 2 ∂ x H(Z i,N t ) -∂ x H( Zi t ) N t exp a H(Z i,N t ) + exp a H( Zi t ) ≤ C (f ) 1 + C (f ) 2 σ 2 x ϕ rc |X i,N t -Xi t | 2 r i t G i t .
	E[ H(Z j,N + ar i t 2 max (γ, 1)	j 1 ,N t ≤E( G( Zj ) H(Z j 2 ,N t 0 )) + 4E + exp a H( Zi √ γ + 1 δ 2 H(Z i,N )], E[ H(Z j 1 ,N t t 0 L µ H( Zj t ) exp a H(Z i,N ) H( Zj 2 t )], E[ H( Zj 1 t ) H( Zj 2 t )], t )
		+2 H( Zi

t ) 2 ] and E[ H( Zj t ) 2 ]. By Cauchy-Schwarz, it is in fact enough to bound E[ H(Z j,N t ) 2 ] and E[ H( Zj t ) 2 ].

Firstt ) H( Zj t ) and E exp 2a H( Zj t ) . By Cauchy-Schwarz, E exp 2a H( Zj t ) H( Zj t ) ≤ E exp 4a H( Zj t ) E H( Zj t ) 2 1/2 . First, we know E( H( Zj t )) is bounded uniformly in time. By Inequation (4.2.11) from Lemma 4.2.6, we have H(z) exp a H(z) ≤ a H(z) + 1 a (e -2) . Then E H( Zj t ) exp a H( Zj t ) is bounded uniformly in time. In particular, each moment of H( Zj t ) is uniformly bounded in time and it is the case for H( Zj t ) 2 . Now, it remains to bound E exp 4a H( s )ds ≤E( G( Zj 0 )) + 4 B -λ t 0 E L µ H( Zj s ) ds,

4t t ) = γX i,N t + β exp a H(Z i,N t ) -γ Xi t + β exp a H( Zi t ) ≤ γX i,N t -γ Xi t exp a H(Z i,N t ) + exp a H( Zi t ) + γ Xi t + β exp a H(Z i,N t ) -exp a H( Zi t ) . t exp a H(Z i,N t ) + exp a H( Zi t )

≤ γr i t exp a H(Z i,N t ) + exp a H( Zi t ) . Thus γ Xi t + β exp a H(Z i,N t ) -exp a H( Zi t ) t ) ≤a 2 max (γ, 1) H(Z i,N t ) -H( Zi t ) exp a H(Z i,N t ) + exp a H( Zi t ) . t ) + H( Zi t ) + α . Thus H(Z i,N t ) -H( Zi t ) ≤ β + t ) . t ) exp a H( Zi t ) . Now1 = 4 γ + a 2 max (γ, 1) β +

  1,i t , I 2,i t , I 3,i t and I 4,i t as follows: Xi t| (-δ + γ + L X (δ + 2) + L C ) + |C i t -Ci t |(L X (δ + 2) + L C ) +σ 2 c ϕ rc |2(X i t -Xi t ) -(C i t -Ci t )| X L X + β X L C ) C L X + β C L C )We then have the additional constraint of δ > 2 (so that the coecient appearing in front of |(X i t ) 3 -( Xi t ) 3 | in the expression of Ki t is non positive). Otherwise, we deal with the various terms exactly as previously, through the choice of a suciently Comportement en temps long de diérents processus stochastiques en neuroscience Résumé: Dans cette thèse, nous nous concentrons sur deux modèles stochastiques pouvant être appliqués aux neurosciences : le modèle de Hawkes et le modèle de FitzHugh-Nagumo. Nous étudions leur comportement en temps long. Le premier chapitre porte sur les processus cumulatifs, qui sont une classe de processus plus généraux que les processus de renouvellement. Ces processus cumulent des variables aléatoires indépendantes au cours du temps. Ces variables aléatoires sont ajoutées sur des intervalles de temps donnés par un processus de renouvellement. En nous inspirant des travaux de Lefevere, Mariani et Zambotti (2011), nous démontrons un Principe de Grandes Déviations pour ces processus, ainsi que des inégalités de grandes déviations dans un cadre plus général. Le second chapitre est dédié aux processus de Hawkes, dans un contexte non linéaire, avec une fonction de reproduction signée. Ils permettent ainsi de modéliser de l'auto-excitation et de l'auto-inhibition. Nous prouvons une loi des grandes nombres, un théorème central limite et des résultats de grandes déviations pour un processus de Hawkes (unique). Ces résultats reposent sur une structure de renouvellement pour ces processus, introduite par Costa, Graham, Marsalle et Tran (2020), qui permettent de dénir les processus de Hawkes comme des processus cumulatifs. Nous utilisons alors des résultats déjà connus pour les processus cumulatifs et les résultats obtenus dans le chapitre 1. Nous exhibons également deux exemples dans lesquels des calculs explicites sont faits. Le dernier chapitre est un travail eectué en collaboration avec Pierre Le Bris et est consacré à l'étude de plusieurs processus de FitzHugh-Nagumo stochastiques en interaction. La spécicité de ce modèle déni par des Equations Diérentielles Stochastiques est son terme cubique dans la dérive, qui est donc non-Lipschitz. Nous nous intéressons au cadre d'interactions champ moyen, et nous montrons une propagation du chaos, d'abord non-uniforme en temps puis uniforme en temps. Pour ce faire, nous utilisons une méthode de couplage mixte, c'est-à-dire un couplage synchrone sur un certain sous-espace et un couplage symétrique sur l'espace complémentaire. Nous exhibons également des bornes explicites pour ces résultats.

	Ki t =G i t 2cf (r i t ) + 2f (r i t )σ 2 c ϕ rc |2(X i t -Xi t ) -(C i t -Ci t )|	2
	+f (r i t ) 2(X i t -Xi t ) -(C i t -Ci t ) (δ + 1) -|(X i t ) 3 -( Xi t ) 3 |(δ -2)
	+|X i t -2	C	(f ) 1 + C	(f ) 2	r i t
	+ f (r i t )	  4 B -	λ 8	H( Zi t ) -	λ 8	H(Z i,N t ) -	λ 8N	N j=1	H( Zj t ) -	λ 32N	N j=1	H(Z j,N t )	  ,
		I 1,i t =G i t f (r i t )	  (δ + 2)	  1 N	N j=1	K X ( Zi	 
														
					+	 1 N			 ,
	I 2,i t =G i t f (r i t )	  (δ + 2)	  L X N	 	N j=1	|X j,N t	-Xj t | + |C j,N t	-Cj t |	   
	+	  L C N	 	N j=1	|X j,N t	-Xj t | + |C j,N t	 t | -Cj 	    -cf (r i t )G i t
	-f (r i t )	λ 32	H( Zi t ) exp a H( Zi t ) +	λ 32	H(Z i,N t ) exp a H(Z i,N t )
	+	λ 32N	N j=1	H( Zj		λ 32N	N j=1	H(Z j,N t ) exp a H(Z j,N t )	  ,
	I 3,i t = f (r i t )	  (α N j=1 |X j,N t | N	2	exp a H(Z i,N t )
					+ (α N j=1 |C j,N t | N	2	exp a H(Z i,N t )
					-	λ 16	H(Z i,N t ) exp a H(Z i,N t ) -	λ 16N	N j=1	H(Z j,N t ) exp a H(Z j,N t )	  .

t -Zj t ) -K X * μt ( Zi t ) N j=1 K C ( Zi t -Zj t ) -K C * μt ( Zi t ) t ) exp a H( Zj t ) +

2.5 The functional I n . . . . . . . . . . . . . . . . . . . . . . . . . 44

1 ).

Acknowledgement. We warmly acknowledge two referees for their careful reading, their constructive comments and their criticism.

Chapter 2. LDP for cumulative processes

Then:

2.B.2 Lower semi-continuity

Lemma 2.B.2. Let E be a closed subset of M 1 ((0, +∞] 2 × {1, ..., n}). Let f be a bounded lower semi-continuous function on (0, +∞] 2 × {1, ..., n}. Let R > 0. Let Φ dened on M 1 ((0, +∞] 2 ) by:

Then Φ is lower semi-continuous.

Proof. We have to prove that for any µ 0 ∈ M 1 ((0, +∞] 2 × {1, ..., n}), for any sequence (µ k ) k such that µ k µ 0 , we have: lim inf Φ(µ k ) ≥ Φ(µ 0 ).

Let show that µ ∈ E → µ(f ) is lower semi-continuous. We remind that for any positive function g on (0, +∞] 2 × {1, ..., n}, we have :

f is bounded: let A such that ∀(x, y), f (x, y) ≥ -A. Let g = f + A, g is a positive lower semi-continuous function.

Let µ 0 ∈ E and (µ k ) k ∈ E such that µ k µ 0 . Then, by Fatou's lemma:

But {(x, y), g(x, y) > t} is an open set because g is lower semi-continuous. By Portmanteau theorem, we have lim inf µ k ({g > t}) ≥ µ 0 ({g > t}). Then:

And we have:

We can nd a neighborhood U of µ 0 included in E c . Then, for all µ ∈ U, Φ(µ) = R. So, for every t < R, for all We also dene for z ∈ R + ,

Similarly we dene Λ * n and J n replacing W 1 by min(W 1 , n). Finally dene

J n (y) .

The latter deviation inequalities are obtained using that J ≤ J (see [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF]). The proof of Theorem 2.4 in [START_REF] Cattiaux | Limit Theorems for Hawkes Processes Including Inhibition[END_REF] is inspired by the one in [START_REF] Lefevere | Large Deviations for Renewal Processes[END_REF] and uses a contraction principle applied to the Large Deviation Principle (LDP for short) at the level of the empirical process. The full LDP is obtained when all exponential moments exist.

Otherwise we mainly obtain the upper bound, which is enough to get asymptotic deviations results, as explained in the Theorem below.

Corollary 3.2.13. Recall that θ 0 is dened in Proposition 3.2.6 (ii).

(

If θ 0 = +∞, N h t /t satises the same LDP as N h t /t.

Chapter 4

Uniform in time propagation of chaos in networks of

FitzHugh-Nagumo neurons

This chapter is a joint work with Pierre Le Bris. It will soon be submitted.

Abstract

In this article, we are interested in the behavior of a fully connected network of N neurons, where N tends to innity. We assume that neurons follow the stochastic FitzHugh-Nagumo model, whose specicity is the non-linearity with a cubic term.

We prove a result of uniform in time propagation of chaos of this model in a meaneld framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle [START_REF] Eberle | Reection Couplings and Contraction Rates for Diusions[END_REF], and recently extended by [START_REF] Durmus | An Elementary Approach to Uniform in Time Propagation of Chaos[END_REF]. Here we consider mean-eld interactions. These interactions are described by two functions K X and K C , applied on the dierence between two states ((X i t , C i t ) -(X j t , C j t )), as described in (4.1.1). In particular, this type of interaction models electrical synapses, which are a sort of synapses.

In [4], Baladron, Fasoli, Faugeras and Touboul study FitzHugh-Nagumo and

Hodgkin-Huxley models with mean-eld interaction, only on X (i.e. K C = 0).

They consider more general interactions, not only applied on the dierence between two states, modeling chemical synapses and electrical synapses. For the FitzHugh-Nagumo model, they consider a noise on X. They prove propagation of chaos, i.e. the convergence of the law of k neurons towards the law of k independent solutions of the mean-eld equations. This article is completed and claried by the work of Bossy, Faugeras and Talay in [START_REF] Bossy | Clarication and Complement to Mean-Field Description and Propagation of Chaos in Networks of HodgkinHuxley and FitzHughNagumo Neurons[END_REF].

Mischler, Quininao and Touboul consider a FitzHugh-Nagumo model in [START_REF] Mischler | On a Kinetic FitzHugh-Nagumo Model of Neuronal Network[END_REF], with a linear interaction on X, and a noise only on X, i.e. σ C = 0 and K X (x, c) = λx.

The drift on X is not exactly the same as in the model above, but remains a cubic function of X, and is very similar. They work on the properties of a solution of the McKean-Vlasov evolution PDE associated to this model and obtain the uniqueness of a global weak solution. Furthermore, they prove that there exists at least one stationary solution, and when the interaction is small, the stationary solution is unique and exponentially stable. They also exhibit numerical results with open problems, like attractive periodic solution in time. [START_REF] Cao | The Kinetic Fokker-Planck Equation with General Force[END_REF] completes their work on the properties of the solution. In a closed framework, in [START_REF] Luçon | Periodicity Induced by Noise and Interaction in the Kinetic Mean-Field FitzHugh-Nagumo Model[END_REF], Luçon and Poquet study the macroscopic limit of a mean-eld model, where they introduce a slow-fast analysis by considering a slow deterministic part, and a fast stochastic and interaction part. In particular, they focus on the periodicity of such a system. They analyze the inuence of both noise and interaction on the emergence of periodic behavior, and prove the existence of periodic solution, exponentially attractive, when the parameters satisfy some assumptions and the drift is "small" enough with respect to interaction and noise.

This model can be made more complex, by considering non mean-eld interaction. In particular, Bayrak, Hövel and Vuksanovi¢ work on a stochastic FitzHugh-Nagumo model with a network interaction in [START_REF] Eyma Bayrak | Modeling Functional Connectivity on Empirical and Randomized Structural Brain Networks[END_REF]. Their type of interaction take into account a connectivity coecient between two neurons, and a propagation velocity.

Other authors choose to consider stochastic FitzHugh-Nagumo with a spatial

Control of the usual distances

The usual metrics, the L 1 distance and L 2 distance, are neither concave nor bounded. In consequence, we work with a semimetric f (r(z, z )) between two particles z ∈ R 2 and z ∈ R 2 , where f has already been dened and r is dened as followed

Let's notice that this semimetric is bounded and concave.

Then, in order to control the behavior of the particles, we use the structure of our systems, and consider the Lyapunov function of our particles. Thanks to this function, we will control the part of second order in the system. To study the distance between two particles systems, we work with ρ((z j , z j ) 1≤j≤N ) dened as followed:

where for each i ∈ {1, ..., N },

An immediate corollary of the denition and properties of H is that ρ is a quantity on R 4N which controls the usual L 1 and L2 distances. In particular, (i) and (iii) will be used to control W 1 .

The proof of this lemma is postponed to Appendix 4.A.4.

Proof of Theorem 4.1.4 in the case σ X > 0

Let ξ > 0 be a parameter destined to vanish, and let ϕ sc : R + → R + and ϕ rc : R + → R + be two Lipschitz continuous functions such that ∀x,

A.1) so that, in particular, for any constant B > 0, if r(z, z

Hence the result:

This inequation yields to (4.2.5) by applying the link between L µemp and L i,N hi (z 1 , . . . , z N ) described in (4.2.1). Let's have (z i ) 1≤i≤N ∈ R 2N and denote H : (z 1 , . . . , z N ) → H(z i ). We dene

we obtain (4.2.5).

First consequences

Proof of Proposition

The second inequality (4.2.7) is obtained exactly the same way.

Chapter 4. Propagation of chaos for FHN neurons close to 0, and it was thus necessary for σ X to be positive (σ C however did not hold any importance). In the case σ X = 0, we then have to assume σ C > 0, and we do a change of variable, motivated by the following observation. We have, when σ X = 0