ticulièrement contribué à mon accueil au LMB, en 2006. Grâce à vous, qui avez assumé les reponsabilités du master modélisation statistique, à l'époque reposant sur les épaules d'une équipe de statisticiens réduite à 3 membres, j'ai pu prendre mes marques sans aucun stress, pour trouver, à mon rythme, ma place dans l'équipe, et dans le LMB. Je vous en suis extrêmement reconnaissant.

Me voici donc à l'étape de l'habilitation, moment important de la vie d'un chercheur. On m'avait dit : "prends-toi un an à l'avance pour constituer ton jury. Beaucoup de professeurs risquent de refuser, ils ne veulent pas perdre de temps à cela". J'ai donc pris contact dès l'année 2013. Vous

La rédaction d'un manuscrit de synthèse est l'occasion parfaite d'écrire noir sur blanc mes remerciement à tous ceux qui auront joué un rôle très important dans mon début de carrière professionnelle.

Mon premier remerciement revient à Philippe Berthet. Je me garderai toujours en mon souvenir ton cours de DEA concernant les processus empiriques, que j'ai eu la chance de suivre, et qui a provoqué en moi un véritable déclic, voire même un électrochoc. Si ma vocation vers la recherche s'était déclarée dès le début de mes études supérieures, elle ne prit de forme concrète qu'à ce moment précis. J'ai pu faire mes premiers pas en recherche avec toi, pendant que tu encadrais mon mémoire de DEA. Je l'ai fait avec une intensité fortement attribuable à l'enthousiasme contagieux dont tu fais preuve à chaque conversation de maths. Avec le recul des années, je me rends compte que le sujet de mémoire que tu m'avais proposé était d'une profondeur idéale pour maintenir la stimulation de l'étudiant que j'étais, en parsemant le parcours de nouvelles étincelles, au gré des articles, dans lesquels j'ai pu me plonger avec passion. Ce mémoire de DEA fut pour moi un événement majeur de ma carrière de chercheur. Grâce à toi j'ai été mis sur de bons et solides rails.

Mon second remerciement est directement adressé à l'ensemble du Laboratoire de Mathématiques de Besançon. Il m'est superflu de parler de son niveau scientifique, l'AERES s'en est chargée elle même. Mais, quitte à choquer certains, ce n'est pas l'excellence que je remercie ici : c'est l'ambiance. Une ambiance de travail avant tout sereine et conviviale. Avec bien sûr ses couacs occasionnels, mais somme toute bien peu fréquents. Combien d'autres laboratoires pourraient rivaliser avec le LMB sur cet aspect, qui a toute son importance pour des activités d'enseignement et de recherche de qualité ? S'il fallait une note, je mettrais A + sans hésiter.

Une mention particulière à vous deux, Jean-Yves Dauxois The empirical processes theory, and its applications to statistics, have been the core of my research works. In the two following subsections I will give a definition of two central objects, each of them having connections with distinct subfields of nonparametric statistics.

The general empirical process

Consider a measurable space X, X . Write M for the set of all probability measures on X, X . Given P ∈ M, and g ∈ L 1 (P), we shall write

P(g) := X gdP.
The general empirical process is defined as follows.

-Given a probability space (Ω, A, P), consider an independent, identically distributed [i.i.d.] sequence of random variables (Z i ) i≥1 , from (Ω, A, P) to X, X . We will denote by P 0 ∈ M the common law of each Z i . -For fixed n ≥ 1, define the empirical measure

P n := 1 n n i=1 δ Z i ,
as a (composition) map from Ω to M. -Consider a class of functions G ⊂ L 1 (P 0 ). Given n ≥ 1, define the corresponding empirical process by G n (•) := g → √ n P n (g) -P 0 (g) , as a map from Ω to R G , where the latter is the space of all real functions on G.

CHAPTER 1. INTRODUCTION

Note that P n is random, for each n ≥ 1, while P 0 is not.

In statistics, G n has a particular relevance, since its stochastic behavior provides a description of the random spatial setting of the sample (Z 1 , . . . , Z n ), which, itself, rules the stochastic behavior of general estimating procedures such as , e.g., Z-estimation, M -estimation, bootstrap procedures, or empirical likelihood procedures (see, [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF]Part 3] for a large, yet non exhaustive overview of the possibilities).

The local empirical process

A particular domain of nonparametric statistics concerns smoothing techniques, toward the estimation of Lebesgue densities or conditional expectations. In this framework, it is of interest to understand the stochastic behavior of local empirical processes. Taking X := R d and a class of functions G ⊂ L 1 (P 0 ), we can define, for n ≥ 1, z ∈ R d , h > 0, and g ∈ G :

T n (g, h, z) := n i=1 g h,z (Z i ) -E g h,z (Z i ) , where g h,z (z) :=g h -1 z -z , z ∈ R d .
This object describes the random setting of the sample locally around z, with h > 0 (usually called the bandwidth) playing the role of the localization strength. Given a class of functions G, the asymptotic behavior, as n → ∞ together with h → 0, of T n (•, h, z) (as processes indexed by G) has been intensively investigated in the literature over the past three decades (an illustrative, yet non exhaustive list of references is [START_REF] Berthet | On the rate of clustering to the Strassen set for increments of the empirical process[END_REF][START_REF] Berthet | Inner rates of coverage of Strassen type sets by increments of the uniform empirical and quantile processes[END_REF][START_REF] Berthet | Some exact rates in the functional law of the iterated logarithm[END_REF][START_REF] Deheuvels | Chung type functional laws of the iterated logarithm for tail empirical processes[END_REF][START_REF] Deheuvels | Strong approximation of quantile process by iterated Kiefer processes[END_REF][START_REF] Deheuvels | Probabilities of hiting of shifted small balls by centered Poisson processes[END_REF][START_REF] Deheuvels | Nonstandard functional laws of the iterated logarithm for tail empirical and quantile processes[END_REF][START_REF] Deheuvels | Functional laws of the iterated logarithm for the increments of empirical and quantile processes[END_REF][START_REF] Deheuvels | Functional laws of the iterated logarithm for local empirical processes indexed by sets[END_REF][START_REF] Deheuvels | Nonstandard local empirical processes indexed by sets[END_REF][START_REF] Einmahl | The as behavior of the weighted empirical process and the lil for the weighted tail empirical process[END_REF][START_REF] Einmahl | Poisson and gaussian approximation of weighted local empirical processes[END_REF][START_REF] Einmahl | Gaussian approximation of local empirical processes indexed by functions[END_REF][START_REF] Mason | A strong invariance principle for the tail empirical process[END_REF][START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF][START_REF] Mason | A refinement of the KMT inequality for the uniform empirical process[END_REF]). It has to be noted that, the relationship between the T n (•, h, z) and nonparametric estimation should not be narrowed to the Parzen-Rosenblatt and Nadaraya-Watson estimators. For example, estimation by wavelet expansions or by local polynomials are also linked to T n (•, h, z), and more generally, this is also the case for local M -estimation or Z-estimation, bootstrap procedures, or empirical likelihood procedures.

Contributions to the study of the general empirical process

In this section, I will sum up my contributions to the general empirical processes theory.

The uniform entropy condition

An important result in empirical processes theory is that a class of functions G having a square integrable envelope and admitting a finite uniform entropy integral is Donsker. That condition, very commonly encountered in practice, has the particularity of involving suprema of packing numbers over all possible L 2 (Q)-norms induced by probability measures Q. I

CONTRIBUTIONS TO THE STUDY OF THE GENERAL EMPIRICAL PROCESS

investigated how these uniform bounds could be used as a basis for proving Donsker and Glivenko-Cantelli theorems for sequences of random measures of the form

P n := i≥1 β i,n δ Z i,n .
The initial motivation of this work came from Bayesian nonparametric theory, where discrete priors and posteriors can be represented this way, most particularly the Dirichlet process prior/posterior random measure.

-Contribution [START_REF] Varron | Donsker and glivenko-cantelli theorems for a class of processes generalizing the empirical process[END_REF]. I showed that the finite uniform entropy integral condition still does adapt very well to the more general form of P n , when the (Z i,n ) i≥1 are conditionally i.i.d given (β i,n ) i≥1 . Indeed, that uniform entropy numbers condition turns out to be sufficient (beside a minimal integrability condition on the envelope of G) to prove a Glivenko-Cantelli and a Donsker theorem, as soon as the conditional distributions of Z 1,n given (β i,n ) i≥1 converge to a limit in the sense of Sheehy and Wellner [START_REF] Sheehy | Uniform donsker classes of functions[END_REF]. Note that boostraped empirical procedures fall into this framework (see, e.g., [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF]Chapter 3.6]). A direct consequence of that result is an alternate proof of Doob's theorem and Bernstein-Von Mises theorem due to James [START_REF] James | Large sample asymptotics for the two-parameter poisson-dirichlet process[END_REF], under the topology spanned by || • || G , for posterior distributions of the Dirichlet process prior.

Contributions to empirical likelihood theory

Empirical likelihood is a nonparametric method of building confidence regions, which was developed by Owen [START_REF] Owen | Empirical Likelihood[END_REF]. It has many advantages over the usual asymptotic gaussian pivot, and turns out to be a serious competitor to the pivotal bootstrap.

-Contribution [START_REF] Dauxois | Non-parametric Tests for Recurrent Events under Competing Risks[END_REF], [START_REF] Varron | Simultaneous confidence bands for some functional plug-in parameters : a computationally feasible approach[END_REF]. With J-Y Dauxois and A. Flesch, we investigated the estimation of functional parameters by empirical likelihood, in a particular setup of lifetime data analysis. The parameter of interest was the joint trajectories of the mean numbers of recurrent events under competing risks, with independent right censoring, and in the presence of a terminal event. These works led to investigating a more general framework, where the functional parameter of interest can be expressed as T (P n ), with T fulfilling some Gâteau and Hadamard differentiability conditions, relatively to norms of the form || • || G . These works are improvements of already existing results of Bertail [START_REF] Bertail | Empirical likelihood in some semiparametric models[END_REF] and [START_REF] Hjort | Extending the scope of empirical likelihood[END_REF], to multisample, multivariate settings, and with an additional care on practical implementation.

Contributions to associated discrete kernel estimation

Assume that the law of Z 1 is discrete with support T. The nonparametric estimation of the probability mass function (p.m.f) f (•) by the empirical p.m.f.

f n : z → P n ({z}) CHAPTER 1. INTRODUCTION
can be put into the wider framework of discrete associated kernel estimators

f n,h : z → 1 n n i=1 K z,h (Z i )
where K = K z,h (•), z ∈ T, h ≥ 0 is a family of p.m.f on T. Despite its expression, f n,h has nothing to do with the local empirical process.

-Contribution [START_REF] Kokonendji | Performance of the discrete associated kernel methods for the total variation distance[END_REF]. With C. Kokonendji, we investigated the consistency of this class of estimators, for the total variation distance

T V (f n,h , f ) := 1 2 T | f n,h (x) -f (x) | dc(x) + 1 2 T f n,h (x)dc(x) -1 ,
where c denotes the counting measure on T, under assumptions on K that are satisfied in practice. We could also derive concentration inequalities for T V (f n,h , f ).

Contributions to the study of local empirical processes

Under an additional condition of uniform boundedness on G, or on integrability of its envelope, both local an general empirical processes are viewed as random elements in ∞ (G), || • || G , where ∞ (G) is the space of all bounded real functions on G, and where

|| ψ || G := sup g∈G | ψ(g) |, ψ ∈ ∞ (G).
One can very rouhgly divide the existing literature into the following types of results.

Asymptotic results

(a) Donsker theorems and weak approximations.

(b) Functional limit laws and strong approximations.

(c) Large deviation principles.

Non asymptotic results

-Concentrations inequalities.

-Control of first moments by chaining techniques.

Concerning the local empirical process, most of my work takes place in (b) and (c). This is the point of the next subsections.

Refinements of existing functional limit laws

Consider a compact set H ⊂ R d , and assume that, for an open set O ⊃ H, the following condition is satisfied : Now write, given n ≥ 1 and h > 0 :

Θ n (h) := T n (•, h, z) r n (h, z) , z ∈ H ,
Where r n (h, z) > 0. A functional limit law for the local empirical process is any theorem stating the almost sure inner and outer topological limits (in

( ∞ (G), || • || G )) of the sequence Θ n (h n ),
for appropriate normalizing sequences r n (h n , z), where h n > 0 is a deterministic bandwidths sequence.

-When H is finite (say, a singleton {z 0 }), those type of results are called local functional limit laws, and r n (h, z 0 ) := 2f (z 0 )nh d log log(n) is the appropriate type of normalization.

-When H has a nonempty interior, those type of results are called global functional limit laws, and r n (h, z) := 2f (z)nh d log(1/h d ) is the appropriate type of normalization. -When the inner/outer topological limits are related to the large deviations properties of the associated Gaussian processes (Strassen-type sets), those limit laws are called standard. This happens when h n tends to zero, but not too fast. For local limit laws, "not too fast" means

nh d n / log log(n) → ∞, (1.1)
while this term has to be understood as

nh d n / log(1/h d n ) → ∞, (1.2)
for global functional limit laws. Standard functional limit laws entail several (strong) consistency or uniform consistency (over a compact H) results for nonparametric or semiparametric estimators. -When the inner/outer topological limits are related to the large deviations properties of the associated Poisson processes, those limit laws are called nonstandard. This happens when the convergence h n is slightly too fast, which means replacing the symbol ∞ by a nonzero constant in (1.1) or (1.2). Nonstandard functional limit laws entail several non consistency or non uniform consistency results (in the strong sense) for estimators.

The simplest form of the local empirical process consists in taking d = 1, Z i uniformly distributed in [0, 1], and G = G 0 as the class of indicators of intervals of the form [0, t], t ∈ [0, 1] d . The object T n (•, h, z) is then called a functional increment of the uniform empirical process. The pioneering works are due to [START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF], [START_REF] Deheuvels | Functional laws of the iterated logarithm for the increments of empirical and quantile processes[END_REF][START_REF] Deheuvels | Nonstandard functional laws of the iterated logarithm for tail empirical and quantile processes[END_REF], proving both standard and nonstandard functional limit laws for those functional increments. Since then, the challenge was to extend those results into the following directions 1. To extend G 0 to more general classes of functions. In particular, to confront the usual conditions of uniform entropy numbers and bracketing numbers with those functional My contributions on this field are -Contribution 1 [START_REF] Varron | Some uniform in bandwidth functional results for the tail uniform empirical and quantile processes[END_REF]. In the local standard functional limit law for the increments of the uniform empirical process, I established that the clustering rates of Θ n (h n ) to the corresponding Strassen set are of order log log(n) -2/3 . This result is not surprising in regard of a results in global standard limit laws established by Berthet [START_REF] Berthet | On the rate of clustering to the Strassen set for increments of the empirical process[END_REF][START_REF] Berthet | Inner rates of coverage of Strassen type sets by increments of the uniform empirical and quantile processes[END_REF]. I also filled a small gap in the Chung-Mogulskii laws for the local empirical process, providing rates of approximation of a certain class of points belonging to the boundary of the Strassen set. -Contribution 2 [START_REF] Varron | Some asymptotic results on density estimators by wavelet projections[END_REF][START_REF] Varron | A nonstandard uniform functional limit law for the increments of the multivariate empirical distribution function[END_REF]. In the global nonstandard functional laws, I managed to attain the multidimensional setting (d ≥ 2), under (Hf ), for any uniformly bounded class G with supports included in a common bounded set [-M, M ] d , admitting finite bracketing numbers (for the uniform distribution on [-M, M ] d ). I pointed out a consequence of that result to the asymptotic behavior of estimators of densities by wavelet projections. -Contribution 3 [START_REF] Maumy | Non standard functional limit laws for the increments of the compound empirical distribution function[END_REF]. With M. Maumy-Bertrand, we proved the local counterpart of Contribution 2, and also extended this local law to the more general object T n,c (g, •, h, z), where

T n,c (g, g, h, z) := n i=1 g h,z (Z i )g(Y i ),
for a function g taking values in R k , for which we made assumptions of finite conditional exponential moments for g(Y 1 ) (note the absence of a centering parameter). Our result was limited to the particulat class G 0,d of indicators of hypercubes of [0, 1] d .

1.3.2

The impact of the in-bandwidth uniformity to the usual functional limit laws

A very central question has animated a significant part of my research works. It can be stated as follows :

What is the impact, on the existing functional limit laws, of requiring an additional in-bandwidth uniformity for the convergences?

More precisely, given two deterministic bandwidth sequences h n ≤ h n , can we still determine the almost sure topological limits of

Θ n := hn≤h≤hn Θ n (h),
and do those limits depend on the widths of the intervals [h n , h n ]?

The original motivation comes from nonparametric statistics, where in-bandwidth-uniform consistency results for nonparametric estimators provides a rigorous justification for the use of data driven bandwidth selection procedures. I provided the following answers.

-Contribution 1 [START_REF] Varron | Some uniform in bandwidth functional results for the tail uniform empirical and quantile processes[END_REF]. When two bandwidths h n ≤ h n have different orders of convergence to 0 (i.e

h n = o(h n ) = o(1)
), an asymptotic independence shows up : for fixed z, T n (•, h n , z) and T n (•, h n , z) obey the same large deviation principle and Strassen law as of two independent Wiener processes. I will call this phenomenon between bandwidths asymptotic independence. That phenomenon was already partially visible in [START_REF] Deheuvels | Asymptotic independence of the local empirical process indexed by functions[END_REF]. -Contribution 2 [START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF]. For local standard functional limit laws (see [START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF][START_REF] Einmahl | Gaussian approximation of local empirical processes indexed by functions[END_REF]), there is an impact of requiring an in-bandwidth uniformity in h ∈ [h n , h n ], which is clearly due to the between bandwidths asymptotic independence. Indeed, when

log log(h n /h n )/ log log(n) → δ ∈ (0, 1], (1.3)
the inner and outer topological limits of T n are not anymore ∅ and S G,1 , but S G,δ and S G,1+δ , where for a ≥ 0, S G,a equals √ a times the Strassen ball of the associated Gaussian process.

-Contribution 3 [START_REF] Varron | A limited in bandwidth uniformity for the functional limit law for the increments of the empirical process[END_REF]. For global standard functional limit laws ( [START_REF] Deheuvels | General asymptotic confidence bands based on kernel-type function estimators[END_REF], [START_REF] Mason | A strong invariance principle for the tail empirical process[END_REF]), there is no impact of the uniformity in h ∈ [h n , h n ]. At this level of normalization, nothing is visible. -Contribution 4 [START_REF] Van Keilegom | Uniform in bandwidth exact rates for a class of kernel estimators[END_REF]. With I. van Keilegom, we investigated the global functional limit law for the more general sets

Θ n,c = T n,c (•, •, h, z) -E T n,c (•, •, h, z) r n (h, z) , h ∈ [h n , h n ], z ∈ H .
We did not prove that functional limit law. However, following initial works of Einmahl and Mason [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF], we proved a closely related result, which is weaker in its form (because it is non functional), but which covers a larger framework in regression estimation.

Asymptotic results in a spatial setting

A very large part of the asymptotic results in empirical processes theory can be roughly stated as "both the empirical process and the associated Gaussian process share the same property". Wschebor [START_REF] Wschebor | Sur les accroissements du processus de Wiener[END_REF] proved that the increments of the wiener process -1/2 W (t + ) -W (t) almost surely satisfy the property

∀B Borel , lim →0 λ t ∈ [0, 1], W (t + ) -W (t) √ ∈ B = µ(B),
where µ is the standard normal distribution and λ stands for the uniform distribution on [0, 1]. This almost sure phenomenon can be roughly called an almost sure spatial convergence in distribution of the increments of the Wiener process.

-Contribution 1 [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF][START_REF] Varron | The almost sure behavior of certain spatial distributions of local empirical processes indexed by functions[END_REF]. I extended that result for the functional increments of the uniform empirical process, proving an almost sure spatial Donsker theorem , toward a Wiener measure. I then extended this result to the general form of T n (•, h, z), under (Hf ), for classes of functions G admitting a finite uniform entropy numbers integral, and for a large class of probability distributions different from λ (including those admitting a square integrable density, and excluding those having atoms). -Contribution 2 [START_REF] Varron | A spatial strassen-type functional limit law for the increments of the local empirical process[END_REF][START_REF] Varron | The almost sure behavior of certain spatial distributions of local empirical processes indexed by functions[END_REF]. In the same vein, I also established the almost sure spatial local standard functional limit law for Θ n . I also proved the nonstandard version of that limit law, but restricted to h n = h n , and for the uniform empirical process.

The probabilistic tools that have been developed

For all the above mentioned contribution, the proof relied on extensions and refinement of already existing tools in empirical process theory. Two of them did involve a significant part of my works.

Poissonization techniques

If the sample size is randomized by a Poisson random variable with expectation n, independent of the sequence (Z i ) i≥1 , then the corresponding poissonized local empirical process

Π n (g, h, z) := ηn i=1 g h,z (Z i ) -E g h,z (Z i ) ,
inherits properties of random Poisson measures, which often turn out to be the nonasymptotic counterparts of asymptotic properties of the T n (•, h, z). This properties of independence are very useful for probability calculus. Hence, an effort has been made to approximate the T n (•, h, z) by the corresponding Π n (•, h, z) during the past decades. For finite H and for very small h n = O(log log(n)/n), this can be achieved by a direct strong approximation [START_REF] Deheuvels | Nonstandard functional laws of the iterated logarithm for tail empirical and quantile processes[END_REF][START_REF] Deheuvels | Nonstandard local empirical processes indexed by sets[END_REF]. Outside this case, it is possible to obtain weaker but nevertheless useful approximations. One of them, initiated by J. Einmhal [START_REF] Einmahl | Multivariate empirical processes[END_REF] and generalized by Deheuvels and Mason [START_REF] Deheuvels | Functional laws of the iterated logarithm for the increments of empirical and quantile processes[END_REF][START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF] provided a bound of the form

P T n (•, •, •) ∈ A ≤ CP Π n (•, •, •) ∈ A ,
as processes indexed by G×]0, h 0 ] × H, where the constant C does not depend on A nor on n. This happens as soon a there is a non null probability that Z 1 falls outside the union of all supports of g h,z , g ∈ G, h ≤ h 0 , z ∈ H. This very useful inequality was put in a wider framework by Giné, Mason and Zaitsev [START_REF] Giné | The L 1 -norm density estimator process[END_REF], for [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF]. I extended inequality (1.4) to a wider class of truncating functions on (X i ) i≥1 , for which the arguments are the partial sums histories up to n (or η n ), namely the vector

P n i=1 1 B (X i )X i ∈ A ≤ CP ηn i=1 1 B (X i )X i ∈ A , as soon as P X 1 / ∈ B > 0. -Contribution 1
X 1 , X 1 + X 2 , . . . , X 1 + . . . + X n .
As an example of application, it is now possible to poissonize maximal inequalities for sums of i.i.d random variables.

Though already very useful, inequalities (1.4) and (1.4) cannot make a connection, between

T n (•, •, •) and Π n (•, •, •)
, of some crucial facts needed for functional limit laws. As a matter of fact, it is not possible to directly translate assertions of the type

n = O P Π n (•, h n , z) ∈ A n
into an assertion of the same type for T n (•, h n , z).

-Contribution 2 [START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF]. I proved an approximation result between T n (•, h n , z) and Π n (•, h n , z), by providing deviation probabilities for large exceedances of

|| T n (•, h n , z) - Π n (•, h n , z) || G .
Those deviation probabilities opened new possibilities of using Poissonization in the local standard functional limit laws. Moreover, these deviation probabilities almost entirely depend on the concentration of η n around n, and thus hold even if η n and (Z i ) i≥1 are not independent, or if η n is not Poisson.

Uniform large deviation principles

Functional limit laws heavily rely on large deviation principles. Arcones [START_REF] Arcones | The large deviation principle of stochastic processes, Part 1[END_REF][START_REF] Arcones | The large deviation principle of stochastic processes, Part 2[END_REF] proposed a criterion to prove large deviation principles in ∞ (G), || • || G . That paradigm decomposes a large deviation principle for a sequence of processes W n into two parts :

1. First, prove a large deviation principle for processes discretized along any finite grid, W n (g 1 ), . . . , W n (g p ) .

2. Second, control their oscillations on the elements of suitable finite partitions.

Thanks to the paradigm of Arcones [START_REF] Arcones | The large deviation principle of stochastic processes, Part 1[END_REF][START_REF] Arcones | The large deviation principle of stochastic processes, Part 2[END_REF], Mason [START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF] showed that it was possible to obtain large deviation principles for T n (•, h, z) without invoking any strong approximation argument.

-Contribution 1 [START_REF] Varron | Some uniform in bandwidth functional results for the tail uniform empirical and quantile processes[END_REF][START_REF] Varron | A limited in bandwidth uniformity for the functional limit law for the increments of the empirical process[END_REF][START_REF] Varron | A spatial strassen-type functional limit law for the increments of the local empirical process[END_REF][START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF]. I investigated how the arguments of Arcones and

1 v n (h) log P * T n (•, h, z) r n (h) ∈ F ≤ -J(F ), for any F closed in ∞ (G), || • || G
, where the presence outer probabilities is due to well known measurability issues for the empirical processes. The uniformity in h ∈ [h n , h n ] turns out to be crucial for proving in-bandwidth-uniform functional limit laws. -Contribution 2 [START_REF] Varron | Une remarque concernant les principes de grandes déviations dans les espaces schauder décomposables[END_REF][START_REF] Varron | A note on weak convergence, large deviations, and the bounded approximation property[END_REF]. The criterion of Arcones can be seen as follows :

1. Given a family of finite rank operators (I α ) α∈Υ of a Banach space (E, || • ||), prove a large deviation principle for I α (W n ) n≥1 for each α ∈ Υ.

Sharply control

|| I α (W n ) -W n || by a suitable choice of α.
I showed that this general idea was rigorously applicable in the context of Schauder decomposable spaces, then more generally in spaces having the bounded approximation property.

Inventory of publications and prepublications

Short notes

The short notes that are preceded by an asterisk have also been the subject of a full lenght article, provided in the next subsection. 2. A spatial Strassen-type functional limit law for the increments of the local empirical process.

3. Simultaneous confidence bands for some functional plug-in parameters : a computationally feasible approach.

4. The almost sure Kuratowski limit of local empirical processes with variable bandwidths 5. (With J-Y. Dauxois and A. Flesch) Empirical likelihood uniform confidence bands in survival analysis under the assumption of competing risks.

6. Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empirical process.

Articles in preparation

1. (With C. Kokonendji) Performances of the discrete associated kernel estimators for the total variation distance.

Chapter 2

Uniform large deviations and Poissonization tools for local empirical processes

This chapter is dedicated to the probabilistic tools that I progressively developed during my research works. In this chapter, and in this chapter only, proofs are given in details. The main motivation of providing such a level of details is that taking time to expose them with precision will allow me to give to be a lot more elusive in the next chapter. Another underlying motivation is that those results somehow reflect the very base of my works : almost every result of Chapter 3 has his proof deeply relying on one of the probabilistic tools of the present chapter.

Uniform large deviation principles

The framework

A large deviation principle (LDP for short) states the rates of convergence to zero of rare events of the form P W n ∈ A , where W n is a sequence of Borel random variables in a topological space. It is an important tool for functional limit laws, because (roughly speaking) it allows to determine the summability or non summability of sequences of such events, and hence apply the Borel-Cantelli Lemma or its converse part. The rates of convergence to zero is determined by a (rate) function J, which describes the stochastic cost of regions of E. Definition 2.1.1 (Good rate function) Let (E, O) be a topological space. A map J : E → [0, ∞] is called a good rate function (or rate function for short) when the sets e ∈ E, J(e) ≤ a , a ≥ 0 are compact sets of (E, O). For A ⊂ E, we shall write J(A) := inf e∈A J(e).

When the topological space is Polish (metric, complete, separable), many criteria have been established to prove large deviation principles (see, e.g. Lynch and Sethuraman [START_REF] Lynch | Large deviations for processes with independent increments[END_REF] for continuous processes with independent increments under the sup-norm, Pukhalskii [START_REF] Pukhalskii | On the theory of large deviations[END_REF]Theorem 2] for Skorokhod spaces, the abstract version of the Gärtner-Ellis theorem [START_REF] Dembo | Large deviations techniques and applications[END_REF]Theorem 4.5.20,p. 157], Bryc's inverse lemma [START_REF] Dembo | Large deviations techniques and applications[END_REF]Chapter 4.4]). Arcones [START_REF] Arcones | The large deviation principle of stochastic processes, Part 1[END_REF][START_REF] Arcones | The large deviation principle of stochastic processes, Part 2[END_REF] was the first to provide such a paradigm in a nonseparable space : the space ∞ (T ), || • || T , where we remind the reader that, for a set T , ∞ (T ) denotes the space of all real bounded functions on T , endowed with the sup norm

|| ψ || T := sup t∈T | ψ(t) | . (2.1)
He showed that a LDP in ∞ (T ), || • || T can always be decomposed into two parts : -For any p ≥ 1 and (t 1 , . . . , t p ) ∈ T p , prove the expected LDP for the finite dimensional random vectors W n (t 1 ), . . . , W n (t p ) . -Prove that the oscillations on a suitable finite partition of T are negligible, in the sense that their probabilities of exceeding arbitrarily small thresholds > 0 tend to zero at sufficiently fast rates.

Arcones makes an explicit use of the particular structure of ∞ (T ), || • || T , by invoking the Arzela-Ascoli theorem as well as extensions of uniformly continuous functions on a set T 0 to its completion. Following a brief remark of Mason [START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF], and after a careful reading of the proofs in [START_REF] Arcones | The large deviation principle of stochastic processes, Part 1[END_REF], I came to the conclusion that all the arguments of Arcones could be extended with no efforts to a larger setup of uniform large deviation principles. The problem of non (Borel) measurability of empirical processes is well known. This is why the following definition uses the concept of inner and outer probability measures (see, e.g. [92, Chapter 1]).

Definition 2.1.2 (Uniform large deviation principle) Let (Ω, A, P) be a probability space and let (W n,ρ ) n≥1, ρ∈Hn be a sequence of collections of maps from Ω to E. Let J be a rate function on (E, O). Let (v n,ρ ) n≥1, ρ∈Hn be a sequence of collections of positive real numbers such that lim

n→∞ inf ρ∈Hn v n,ρ = ∞.
We say that (W n,ρ ) n≥1, ρ∈Hn satisfies the uniform large deviation principle (ULDP) for (v n,ρ ) n≥1, ρ∈Hn and J, when :

-For all F closed in (E, O) we have

lim n→∞ sup ρ∈Hn 1 v n,ρ log P * W n,ρ ∈ F ≤ -J(F ).
-For all O open in (E, O) we have

lim n→∞ inf ρ∈Hn 1 v n,ρ log P * W n,ρ ∈ O ≥ -J(O).
When the H n are singletons, we will rather speak of a large deviation principle (LDP) for a sequence (W n ) n≥1 .

The conclusion of my early works was as follows. For ψ ∈ R 2. (Asymptotic equicontinuity) For each > 0, there exists a finite partition T of T such that, for each T ∈ T , we have

lim n→∞ sup ρ∈Hn 1 v n,ρ log P * sup (t 1 ,t 2 )∈T 2 W n,ρ (t 1 ) -W n,ρ (t 2 ) ≥ ≤ -1/ .
Then (W n,ρ ) n≥1, ρ∈Hn satisfies the uniform large deviation principle (ULDP) for (v n,ρ ) n≥1, ρ∈Hn and the rate function

J : ψ → sup p≥1, t∈T p J t Π t (ψ) .
Before emphasizing the usefulness of such a criterion for the local empirical process, I will briefly open a parenthesis to a work I made in large deviation theory [START_REF] Varron | Une remarque concernant les principes de grandes déviations dans les espaces schauder décomposables[END_REF][START_REF] Varron | A note on weak convergence, large deviations, and the bounded approximation property[END_REF], and which has his foundations in the works of Arcones for ∞ (T ).

Large deviations in spaces having the bounded approximation property

When (E, O) is a Banach space admitting a Schauder basis (or more generally, a Schauder decomposable space), Suquet [START_REF] Suquet | Tightness in Schauder decomposable Banach spaces[END_REF] proved a tightness criterion for sequences of Borel random variables that involves finite dimensional projections on Schauder basis. Following his ideas, and having in mind the similarity between large deviations and weak convergence in ∞ (T ), || • || T , I investigated the possibility of establishing a similar criterion, for large deviations in such spaces. This was achieved by making use of the concept of LD-tightness introduced by Lynch an Sethuraman [START_REF] Lynch | Large deviations for processes with independent increments[END_REF], which is an appropriate tool when (E, O) is Polish. The next step was then to extend those results to the more general case of spaces (E, || • ||) having the bounded approximation property (BAP spaces), for which the definition is as follows.

Definition 2.1.3 (BAP spaces) Let I α α∈Υ be a net of finite rank linear operators from E to E. The space (E, || • ||) is said to satisfy the bounded approximation property (BAP for short) with the net I α α∈Υ when it is a Banach space, and when :

lim α∈Υ sup e∈K || I α (e) -e || for each compact K, sup α∈Υ ||| I α |||< ∞,
where lim α∈Υ stands for the limit along the net Υ, and where :

||| L |||:= sup{| L(x) |, x ∈ E, || x ||= 1} for a linear operator L in E.
Note that this framework encompasses ∞ (T ), || • || T , as well as spaces having a Schauder basis. Any Banach space (E, || • ||) can be identified to a closed subspace of ∞ (T ), || • || T , by taking T as the unit ball of its topological dual. In empirical processes theory, such an identification can lead to wrong tracks, as it is for example the case for the central limit theorem in Banach spaces (see, e.g., [92, p. 92]). The result of my works, however, is that the above-mentioned identification turns out to be an efficient way to extend the results of Arcones to BAP spaces.

Theorem 1 (Varron, 2014 [106]) Let W n n≥1 be a sequence of maps from Ω to E. Consider the following conditions :

-(B1) For each α ∈ Υ, the sequence I α (W n ) n≥1 is Borel measurable, and satisfies the LDP in (E, || • ||) for v n and the rate function J α .

-(B2) For each > 0 we have

lim α∈Υ lim n→∞ 1 v n log P * || W n -I α (W n ) ||> ≤ -1/ .
-(b1) The sequence (W n ) n≥1 satisfies the LDP for the rate function J.

Then (B1), (B2) together imply (b1), with J defined as

J(x) := sup α∈Υ J α I α (x) , x ∈ E.
Moreover, (b1) implies both (B1) and (B2) with

J α (x) := inf J(y), I α (y) = x .
A by product of this result was a functional limit law, in Hölderian topologies, for smoothed versions of the uniform empirical process on the real line, by convolutions with K(h -1 n •), where K is Hölder (see, [START_REF] Varron | A note on weak convergence, large deviations, and the bounded approximation property[END_REF]Theorem 2.4]).

A ULDP for the local empirical process

The extension of the works of Arcones did open several opportunities for my works on functional limit laws for the local empirical process, for which I recall the definition for n ≥ 2.1. UNIFORM LARGE DEVIATION PRINCIPLES 1, z ∈ R d , h > 0, and g ∈ G :

T n (g, h, z) := n i=1 g h,z (Z i ) -E g h,z (Z i ) , where g h,z (z) :=g h -1 z -z , z ∈ R d .
Our assumption on the law of Z 1 is as follows (as indicated in the introduction).

(Hf ) Z 1 admits a Lebesgue density on an open set O for which there exists a version f that is continuous and bounded away from 0 and ∞ on O.

We will then consider a compact set H ⊂ O. Such a regularity condition as (Hf ) allows to have in mind the heuristic (λ standing for the Lebesgue measure)

P Z 1 ∈ z + hJ ∼ λ(J)h d f (z), (2.3)
for bounded J, as h → 0, uniformly in z ∈ H.

The main advantage of Proposition 2.1.1 is that, taking

H n := H × [h n , h n ], it allows to simultaneously control probabilities of rare events of T n (•, h, z), z ∈ H, h ∈ [h n , h n ],
even if the rate of converge to depends on h (for example, it can be of order h J(F ) for closed F ). Each article I wrote or co-wrote on standard functional limit laws involves the proof of an ad-hoc ULDP, which then is used as the very base of asymptotic probability calculus. At the price of very slightly strengthening the assumptions made in those articles, it is possible to unify all of these ULDP into a single one, which I will present and prove here. Before stating our assumptions on G we will recall some definitions that are usual in empirical processes theory.

Definition 2.1.4 (Covering numbers) Let (E, d) be a metric space, and let E 0 ⊂ E be an arbitrary set. For all > 0, we denote by N ( , E 0 , d) the minimal (possibly infinite) number of closed d-balls of radius needed to cover E.

Definition 2.1.5 (Pointwise separable class) A class G of real functions on a set X is said to be pointwise separable when there exists a countable subclass G 0 such that, for each g ∈ G there exists a sequence (g n ) n≥1 ∈ G N 0 such that g n (x) → g(x), as n → ∞, for all x ∈ X.

We will consider a class of functions G satisfying :

(P ointw. sep.) The class G is pointwise separable on R d , (Bounded) There exists M > 0 such that | g |≤ M for each g ∈ G, (Support)
There exists M > 0 such that g(z) = 0 for each g ∈ G and z /

∈ [-M, M ] d , (U nif. entropy) We have ∞ 0 sup Q probab. log N , G, || • || Q,2 d < ∞,
where, for any measure Q and r > 0, the symbol || • || Q,r denotes the corresponding L r (Q)norm. Note that, when both assumptions (Bounded) and (Support) are made simultaneously, I will implicitly use a parameter M > 0 fulfilling both of them. A combination of assumptions (U nif. entropy) and (Support) is largely sufficient to ensure the existence of L 2 (λ) isonormal Gaussian process W G on G, namely a centered gaussian process having covariance function

Cov W G (g 1 ), W G (g 2 ) := [-M,M ] d g 1 g 2 dλ,
for which we can consider the associated rate function

J G : ψ → inf || g || 2 λ,2 , g Borel, ψ(•) ≡ g → R d ggdλ , (2.4)
with the convention inf ∅ = ∞. It is well known that J G rules the large deviation properties of W G (see [28, p. 85]).

In a metric space (E, d) we shall write Our general ULDP can be stated as follows. An indexation in k has been chosen (instead of n), because this ULDP will be systematically used for subsequences. Note that, form now on, unless otherwise specified, each vector space R p will be endowed with its Euclidian norm || • || p .

A := x ∈ E, d(x, A) < , A ⊂ E, > 0, with (2.5) d(x
Proposition 2.1.2 (Varron, 2008(Varron, -2014, from [95, 98, 108], from [95, 98, 108]) Let ñk be a strictly increasing sequence of integers, and let hk , hk , k be three sequences of non negative real numbers such that hk ≤ hk , hk → 0 and k → 0. Let (v k (h)) k≥1,h∈[ hk , hk ] be a sequence of collections of nonnegative real numbers such that

lim k→∞ inf h∈[ hk , hk ] v k (h) = ∞, (2.6)
r k := inf h∈[ hk , hk ] ñk h d v k (h) → ∞. (2.7)
Let G be a class of functions fulfilling (P ointw. sep.), (Bounded), (Support), and (U nif. entropy).

Then, under (Hf ), the following assertions hold :

∀F closed in ∞ (G), || • || G , lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] 1 v k (h) log P * T ñk (•, h, z ) 2f (z)ñ k h d v k (h) ∈ F ≤ -J G (F ); ∀Oopen in ∞ (G), || • || G , lim k→∞ inf z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] 1 v k (h) log P * T ñk (•, h, z ) 2f (z)ñ k h d v k (h) ∈ O ≥ -J G (O).

Some comments on the assumptions

In the next paragraphs, I will decompose the proof into several steps, with a particular emphasis on the role of each of the assumptions made upon G, the bandwidths, and Z 1 . This complete proof can be skipped at first reading, (in that case, the reader can go directly to §2.2) but it is nevertheless recommended to have in mind the following sum up of the different roles of the assumptions :

-The only role of (P ointw. sep.) is to make the suprema of empirical processes measurable.

-Assumption hk → 0 in conjunction with (Support) and (Hf ) makes the covariance structure of the local empirical processes converge to the covariance structure of W G uniformly in z ∈ H (see Lemma 2.1.1 in the sequel). -Assumption (2.7) has to be understood as follows : by the heuristic (2.3), and by (Support), the average number of Z i that play a role in the expression of

T ñk (•, h, z) is roughly ñk h d f (z)λ([-M, M ] d )
This average number can be considered as the "true sample size". Hence v k (h) is required to be large in front of this "sample size", in order to consider deviations sufficiently large to observe Gaussian tails. -Those Gaussian tails are determined by a concentration inequality, initiated by Talagrand (see Fact 2.1.2). Assumption (Bounded) plays its role for the use of that concentration inequality. It also plays a role to use an approximation result due to Zaitsev (see Fact 2.1.1) between finite dimensional marginals of the T ñk (•, h, z) and those of W G . -Such a concentration inequality holds around the expectations of suprema of empirical processes. Assumption (U nif entropy) is essential to control these expectations by the usual chaining argument for Rademacher processes (see also §2.1.5 for a discussion about bracketing conditions). -Allowing z to vary in small balls around z is handled by continuity of f . This additional technicality can be ignored at first reading.

Proof :

Since O is open, k → 0, and by (Support), we can assume without loss of generality

that ∀k ≥ 1, h ∈ [ hk , hk ], || z -z || d ≤ k , z / ∈ O, g h,z (z) = 0. (2.8)

Finite dimensional ULDP by Zaitsev's inequality

Following Proposition 2.1.1 we will first prove point 1 of that proposition. To that end, we will use the powerful approximation result of Zaitsev [START_REF] Yu | On the Gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality[END_REF] for, e.g., sums of independent and bounded random vectors. For p ≥ 1, and for two probability measures P and Q on R p , we shall write :

π(P, Q, ) := sup A Borel max P (A) -Q(A ), Q(A) -P (A ) , for > 0.
Fact 2.1.1 (Consequence of Zaitsev [START_REF] Yu | Estimates of the Levy-Prokhorov distance in the multidimensional central limit theorem for random variables with finite exponential moment[END_REF]) There exists universal constants c 1 , c 2 such that, for any p ≥ 1, n ≥ 1, τ > 0, and any sequence (U i ) i=1,...,n of centered, independent random vectors in R p for which each coordinate is almost surely bounded by τ , we have

∀ > 0, π(P, Q, ) ≤ c 1 p 2 exp - c 2 p 2 τ , (2.9)
where P and Q are respectively the law of n i=1 U i and of its Gaussian analogue.

Assumption (Bounded) plays its first role here, since it entails, for each

k ≥ 1, z ∈ H, || z -z || d ≤ k and h ∈ [ hk , hk ], that the random vectors U i,k,h,z,z := g 1 h,z (Z i ) -E g 1 h,z (Z i ) 2f (z)ñ k h d v k (h) , . . . , g p h,z (Z i ) -E g p h,z (Z i ) 2f (z)ñ k h d v k (h) i=1...,ñ k
have each of their coordinates almost surely bounded by

τ k,h,z,z := 2M 2f (z)ñ k h d v k (h) ,
from where, for arbitrary > 0, taking the notations of Fact 2.1.1 :

π P k,h,z,z , Q k,h,z,z , ≤c 1 p 2 exp - 2f (z)ñ k h d v k (h) 2M c 2 p 2 , ≤c 1 p 2 exp - 2f (z)r k v k (h) 2M c 2 p 2 ,
where r k is defined in (2.7), and where P k,h,z,z and Q k,h,z,z respectively denote the law of ñk i=1 U i,k,h,z,z and of its Gaussian analogue. Hence, since both (2.6) and (2.7) hold, we have, for each > 0 :

lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] 1 v k (h) log π P k,h,z,z , Q k,h,z,z , = -∞.
By standard topological arguments, (2.10) provides sufficiently fast approximations to prove that the sought ULDP for

P k,h,z,z z∈H, ||z -z|| d ≤ k , h∈[ hk , hk ] is the same as that of Q k,h,z,z z∈H, ||z -z|| d ≤ k , h∈[ hk Now, each Q k,h,z,z can be represented as the law of (2v k (h)) -1/2 Σ 1/2 h,z,z Z,
where Z is standard normal on R p , and where

Σ h,z,z [j, j ] := f (z) -1 h -d Cov g j h,z (Z 1 ), g j h,z (Z 1 ) , (j, j ) ∈ {1, . . . , p} 2 .
Now write Σ[j, j ] := g j , g j dλ for (j, j ) ∈ {1, . . . , p}. By standard analysis for Gaussian distributions, if we prove that

lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] || Σ h,z,z -Σ || p 2 → 0, (2.10) then Q k,h,z,z z∈H, ||z -z|| d ≤ k , h∈[ hk , hk ]
will satisfy the ULDP for the rate function J g 1 ,...,gp which is defined as the quadratic form on R p associated to Σ. Now (2.10) is a consequence of the following crucial remark.

Lemma 2.1.1 We have

lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] sup (g,g )∈G 2 1 f (z)h d Cov g h,z (Z 1 ), g h,z (Z 1 ) -gg dλ = 0, lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] sup (g,g )∈G 2 1 f (z)h d E g h,z (Z 1 )g h,z (Z 1 ) -gg dλ = 0.
Proof : We use the change of variable u := z +hv in the next calculus for arbitrary (g, g ) ∈ G 2 .

Cov g h,z (Z 1 ), g h,z (Z 1 ) = O g h,z (u)g h,z (u)f (u)du - O g h,z (u)f (u)du × O g h,z (u)f (u)du, by (2.8) =h d f (z) [-M,M ] d g(v)g (v) f (z + hv) f (z) dv -h d f (z) 2 [-M,M ] d g(v) f (z + hv) f (z) dv × h d [-M,M ] d g (v) f (z + hv) f (z) dv, (2.11)
where assumption (Support) plays its role here, since it makes possible to integrate on [-M, M ] d . Then assumption (Hf ) allows the following crucial argument

lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] sup v∈[-M,M ] d f (z + hv) f (z) -1 = 0, (2.12)
and (2.11) is always negligible in front of the first term because h ≤ hk → 0.

Local representation and concentration inequalities

A well understood heuristic about the local uniform empirical process is that, when looking at the spatial repartition of a uniform sample restricted to a small interval [t, t + h n ], with h n "small", the quantity of observed points is of order nh n . This is roughly speaking the true observed sample size. For the local empirical process, the same heuristic is appealing, because, by assumption (Support), the support of any function g h,z is contained in

A(h, z) := z + [-M h, M h] d , z ∈ O, h > 0.
(2.13)

Hence, the observations falling outside this small ball do not play any role in the value of T ñk (•, h, z). Einmahl and Mason [START_REF] Einmahl | Gaussian approximation of local empirical processes indexed by functions[END_REF] gave a form to this heuristic by noticing the following equality in distributions (fo which we will use the symbol = d ) holds for collections of G-indexed processes, for any

k ≥ 1, h > 0 and z ∈ H k , g h,z (Z 1 ), . . . , g h,z (Z ñk ) g∈G = d τ (h,z ) 1 g Y (h,z ) 1 , . . . , τ (h,z ) ñk g Y (h,z ) ñk g∈G , (2.14)
where -Y (h,z ) i≥1 is an i.i.d. sequence having the distribution

P (h,z ) 0 (•) := P h -1 (Z 1 -z ) ∈ • Z 1 ∈ A(h, z ) . (2.15) -τ (h,z ) i i≥1 is an i.i.d. Bernoulli sequence, independent of Y (h,z ) i i≥1
, and having expectation a(h, z ), with a(h, z)

:= P Z 1 ∈ A h,z , z ∈ O, h > 0. (2.16)
Assumption (Support) plays an essential role for that representation, for which a consequence intensively used by Einmahl and Mason [START_REF] Einmahl | Gaussian approximation of local empirical processes indexed by functions[END_REF] is that

T ñk (•, h, z ) := d b k,h,z i=1 g Y (h,z ) i -E g Y (h,z ) i + π k,h,z E g Y (h,z ) i g∈G , (2.17)
where

-b k,h,z is a binomial (ñ k , a(h, z )) random variable, independent of (Y (h,z ) i ) i≥1 , with -π k,h,z is defined as (b k,h,z -ñk a(h, z ))
Hence, at the price of a randomness of the sample size, of modifying the law of the sample, and up to a remaining term that can be controlled by usual inequalities for binomial distributions, the local empirical process can be represented as a general empirical process indexed by the same class of functions G. The symmetrizations techniques are very useful in empirical processes theory. Another consequence of (2.14) is the following representation for the symmetrized versions of the local

2.1. UNIFORM LARGE DEVIATION PRINCIPLES empirical process ñk i=1 σ i g h,z (Z i ) g∈G = d b k,h,z i=1 σ i g Y (h,z ) i g∈G , (2.18)
where, on each side of the representation, (σ i ) i≥1 denotes an i.i.d Rademacher sequence (P

σ 1 = 1 = P σ 1 = -1 = 1/2
∀ψ ∈ ∞ (G), sup p≥1, g=(g 1 ,...,gp)∈G p J g Π g (ψ) = J G (ψ).
Hence, it only remains to prove the (Asymptotic equicontinuity) part of Proposition 2.1.1. This has to be done through the uniform control of the oscillations of the T ñk (•, h, z) on the elements of a suitable finite partition of T = G. By (U nif. entropy) and (Support), G is totally bounded for the norm || • || λ,2 . Hence it is sufficient to prove that, for any > 0, we have

lim δ→0 lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] 1 v k (h) log P sup (g,g )∈G 2 , ||g-g || λ,2 ≤δ T ñk (g , h, z ) -T ñk (g, h, z ) 2f (z)ñ k h d v k (h) > = -∞.
Note that the preceding inequality involves probabilities (instead of outer probabilities) since (P ointw. sep.) is satisfied.

Clearly, (2.19) involves suprema of empirical processes, for which powerful concentration inequalities around their expectations have been established, first by Talagrand [START_REF] Talagrand | Sharper bounds for Gaussian and empirical processes[END_REF], and then refined by Bousquet [START_REF] Bousquet | A Bennett concentration inequality and its application to suprema of empirical processes[END_REF] and Klein and Rio [START_REF] Klein | Une inégalité de concentration à gauche pour les processus empiriques[END_REF]. On the other hand, assumptions (Bounded) and (U nif. entropy), combined with representation (2.18) will provide sharp enough bounds for the expectations of these suprema. The following concentration inequality is due to Talagrand [START_REF] Talagrand | New Gaussian estimates for enlarged balls[END_REF], and then improved by Einmahl and Mason to a maximal inequality [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF].

Fact 2.1.2 (Talagrand, Einmahl, Mason [START_REF] Talagrand | New Gaussian estimates for enlarged balls[END_REF][START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF]) Let G be a class of measurable real functions on a measurable space (X, X ), fulfilling (P ointw. sep.) and (Bounded) for some 0 < M < ∞. Let (Y i ) i≥1 be an i.i.d. sequence of random variables on X, X and (σ i ) i≥1 an i.i.d Rademacher sequence independent of the first sequence. Then, for suitable universal constants A 1 , A 2 > 0 we have, for each t > 0 and n ≥ 1 :

P max m=1,...,n sup g∈G m i=1 g(Y i ) -E g(Y i ) ≥ A 1 E sup g∈G | n i=1 σ i g(Y i ) | + t ≤2 exp - A 2 t 2 nσ 2 G + exp - A 2 t M , with
σ 2 G := sup g∈G Var g(Y 1 ) .
That powerful inequality has to be accompanied by a control of first moment. In our particular setup, the local representation (2.18), combined with Dudley's chaining argument for subgaussian processes leads to a simple bound for those expectations. This is the aim of the following lemma.

Lemma 2.1.2 (Varron, from [START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF]) There exists a universal constant C 0 such that, for all k ≥ 1, z ∈ O, h > 0, and for all class of function G fulfilling (P ointw. sep.), we have

E sup g∈G ñk i=1 σ i g h,z (Z i ) ≤ C 0 ∞ 0 sup Q probab. log N , G, || • || Q,2 d × ñk a(h, z). (2.

19)

Proof : The proof uses arguments that are very similar to those of Dony and Einmahl [START_REF] Dony | Uniform in bandwidth consistency of kernel regression estimators at a fixed point[END_REF]. Assume without loss of generality that the RHS of (2.19) is finite. We first remark that, by subgaussianity of Rademacher processes (see, e.g., [92, p. 127-128]), we have for all

n ≥ 1, z ∈ O, h > 0 : E sup g∈G n i=1 σ i g Y (h,z) i =E Y (h,z) 1 ,...,Y (h,z) n E σ 1 ,...,σn sup g∈G n i=1 σ i g Y (h,z) i ≤E Y (h,z) 1 ,...,Y (h,z) n C 0 √ n ∞ 0 log N , G, || • || P (h,z) n ,2 d , ≤C 0 √ n ∞ 0 sup Q probab. log N , G, || • || Q,2 d ,
where

P (h,z) n = n -1 n i=1 δ Y (h,z) i
, and where C 0 denote a universal constant. Next, using representation (2.18) we have, for all k ≥ 1, z ∈ O, h > 0 :

E sup g∈G ñk i=1 σ i g h,z (Z i ) =E sup g∈G b k,h,z i=1 σ i g(Y h,z i ) ≤ ñk a(h, z)C 0 ∞ 0 sup Q probab. log N , G, || • || Q,2 d . (2.20)
The next Proposition is a combination of the two preceding ones. It is a refinement of existing arguments in the literature [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF][START_REF] Einmahl | Uniform in bandwidth consistency of variable bandwidth kernel estimators[END_REF][START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF]. Its contribution consists in giving precise threshold conditions to give a sense to the heuristics "h has to be small and nh d has to be large to observe Gaussian tails". I also tried properly determine which parameters of the framework (the law of Z 1 and the structural assumptions on G) do determine those thresholds.

UNIFORM LARGE DEVIATION PRINCIPLES

Proposition 2.1.3 (Varron, from [START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF]) Let (Z i ) i≥1 be an i.i.d. sequence for which the law of Z 1 fulfills (Hf ) for some open set O. There exist universal constants C 0 , C 1 , C 2 > 0 such that the following assertion is true : Given a compact set H ⊂ O, and given C 3 < ∞ and M > 0, there exists h 0 > 0 and 0 > 0 depending only on M ,f and H such that, for any pointwise separable class of functions G fulfilling

∀g ∈ G, | g |≤ M 1 [-M,M ] d , (2.21)
∞ 0 sup Q probab. log N ( , G, || • || Q,2 )d ≤ C 3 , (2.22
)

there exists K 0 > 0 such that, for each h > 0, u > 0, n ≥ 1, z ∈ H and z ∈ R d fulfilling h ≤ h 0 , u > (C 0 C 3 ) 2 2 d+1 M d , nh d > K 0 u and || z -z || d ≤ 0 , we have : P max m=1,...,n sup g∈G m i=1 g h,z (Z i ) -E g h,z (Z i ) > C 1 f (z)nh d u ≤ 4 exp - C 2 u ∆ 2 G , where ∆ 2 G := sup g∈G || g || 2 λ,2 . (2.23)
Proof : Fix M > 0. By (Hf ) we can choose h 0 > 0 and 0 > 0 such that

sup z∈H, ||v|| d ≤M h 0 , ||z -z|| d ≤ 0 f (z + v) f (z) ≤ 2. (2.24)
Then, by the same arguments as those used in the proof of Lemma 2.1.1 we have, for any G fulfilling (2.21) :

sup 0<h≤h 0 sup z∈H sup ||z -z|| d ≤ 0 sup g∈G 1 f (z)h d Var g h,z (Z 1 ) ≤ 2∆ 2 G , (2.25)
sup 0<h≤h 0 sup z∈H sup ||z -z|| d ≤h 0 1 f (z)h d a(h, z ) ≤ 2(2M ) d . ( 2
C 0 C 3 na(h, z ) ≤ f (z)nh d u, for all u ≥ (C 0 C 3 ) 2 2 d+1 M d . (2.27) Moreover, the class G(h, z ) := g h,z , g ∈ G ,
is uniformly bounded by M , and satisfies

σ 2 G(h,z ) ≤ 2f (z)h d ∆ 2 G , by (2.

25). Now, for all

h ≤ h 0 , u ≥ (C 0 C 3 ) 2 2 d+1 M d , || z -z || d ≤ 0 ,
g h,z (Z i ) -E g h,z (Z i ) > 2A 1 f (z)nh d u ≤2 exp - A 2 f (z)nh d u nσ 2 G(h,z ) + exp - A 2 f (z)nh d u M ≤4 exp - A 2 u 2∆ 2 G
, as soon as

nh d u ≥ M 2 4∆ 4 G inf z∈H f (z) =: K 0 . (2.28)
This concludes the proof of Proposition 2.1.3.

Asymptotic equicontinuity

Recall that we aim to prove that, for any > 0, we have

lim δ→0 lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] 1 v k (h) log P sup (g,g )∈G 2 , ||g-g || λ,2 ≤δ T ñk (g , h, z ) -T ñk (g, h, z ) 2f (z)ñ k h d v k (h) > = -∞.
(2.29) We use the notations of Proposition 2.1.3. First, note that, by (Bounded) and (U nif. entropy), and by standard covering numbers arguments, all the classes

G δ := g -g , || g -g || λ,2 ≤ δ , δ > 0, (2.30
z ∈ H, || z -z || d ≤ k , h ∈ [ hk , hk ] : k ≤ 0 , h ≤ h 0 , v k (h) ≥ (C 0 C 3 ) 2 2 d+1 M d , and || z -z || d ≤ h 0 , which implies, by Proposition 2.1.3 P sup (g,g )∈G 2 , ||g-g || λ,2 ≤δ T ñk (g , h, z ) -T ñk (g, h, z ) 2f (z)ñ k h d v k (h) > =P sup g∈G δ ñk i=1 g h,z (Z i ) > 2f (z)ñ k h d v k (h) ≤4 exp - 2A 2 2 v k (h) 2∆ 2 G δ . Since ∆ 2 G δ = δ 2 ,
this proves (2.29) and hence concludes the proof of Proposition 2.1.2.

Asymptotic independence of two local empirical processes at very different scales

In this subsection, I will give a more precise meaning to the between bandwidths asymptotic independence phenomenon which I evoked in the introduction (see, §1.3.2). When investigating the large deviation properties of two independent copies

(W 1 G , W 2 G ) of W G (recall (2.4)), the appropriate rate function on ∞ (G) × ∞ (G) is the following : J ⊗2 G (ψ 1 , ψ 2 ) := J G (ψ 1 ) + J G (ψ 2 ), (ψ 1 , ψ 2 ) ∈ ∞ (G) × ∞ (G), (2.31)
where the presence of a summation should not be surprising, as large deviations deal with logarithm of probabilities. The following ULDP is an unpublished extension of a result I published in [START_REF] Varron | Some uniform in bandwidth functional results for the tail uniform empirical and quantile processes[END_REF] for the local uniform empirical process.

( hk ) ∼ v k ( hk ) as k → ∞, we have : ∀F closed in ∞ (G), || • || G × ∞ (G), || • || G , lim k→∞ sup z∈H, ||z -z|| d ≤ k 1 v k ( hk ) log P * T ñk (•, hk , z ) 2f (z)ñ k hd k v k ( hk ) , T ñk (•, hk , z ) 2f (z)ñ k hd k v k ( hk ) ∈ F ≤ -J ⊗2 G (F ); ∀Oopen in ∞ (G), || • || G × ∞ (G), || • || G , lim k→∞ inf z∈H, ||z -z|| d ≤ k 1 v k ( hk ) log P * T ñk (•, hk , z ) 2f (z)ñ k hd k v k ( hk ) , T ñk (•, hk , z ) 2f (z)ñ k hd k v k ( hk ) ∈ O ≥ -J ⊗2 G (O).
Proof : The proof of Proposition 2.1.2 already established the (Asymptotic equicontinuity) criterion uniformly in h ∈ [ hk , hk ], so it remains valid when requiring only simultaneity in { hk , hk }. Hence, we need only to prove the analogue of

(F inite dimensional U LDP ) in ∞ (G) × ∞ (G).
Taking arbitrary p ≥ 1 and (g 1 , . . . , g p ) ∈ G p , the same remark as in the first sentence of the proof holds for the approximation of the law of

S k,z,z := T ñk (g 1 , hk , z ) 2f (z)ñ k hd k v k ( hk ) , . . . , T ñk (g p , hk , z ) 2f (z)ñ k hd k v k ( hk ) , T ñk (g 1 , hk , z ) 2f (z)ñ k hd k v k ( hk ) , . . . , T ñk (g p , hk , z ) 2f (z)ñ k hd k v k ( hk )
by their gaussian analogues on R 2p . Finally, since v k ( hk ) ∼ v k ( hk ), the only assertion to prove is that the covariance matrix of v k ( hk ) S k,z,z converges uniformly in z ∈ H, || z -z || d ≤ k to a block diagonal matrix having two blocks of size p × p, which is a consequence of the following assertion

∀(g, g ) ∈ G 2 , lim k→∞ sup z∈H, ||z -z|| d ≤ k 1 f (z) hd k hd k Cov g hk ,z (Z 1 ), g hk ,z (Z 1 ) = 0. ( 2
lim k→∞ sup z∈H, ||z -z|| d ≤ k 1 f (z) hd k hd k E g hk ,z (Z 1 ) E g hk ,z (Z 1 ) = lim k→∞ hd k hd k sup z∈H, ||z -z|| d ≤ k f (z) 1 f (z) hd k E g hk ,z (Z 1 ) × 1 f (z) hd k E g hk ,z (Z 1 ) =0.
On the other hand we have, for each

k ≥ 1, z ∈ H, || z -z || d ≤ k , 1 f (z) hd k hd k E g hk ,z (Z 1 )g hk ,z (Z 1 ) = hd k f (z) hd k hd k g(u)g hk hk u f (z + hk u)du = hd k hd k M 2 (2M ) d sup z∈H, ||z -z|| d ≤ k , u∈[-M,M ] d f (z + hk u) f (z) ,
where we used both (Bounded) and (Support) in the last inequality. Now, since hk = o( hk ), and by (2.24), that last bound tends to 0, which concludes the proof of (2.32) and also concludes the proof of Proposition 2.1.4.

Some words on bracketing

Another structural assumption frequently made on a class of functions is that it admits a finite bracketing numbers integral, namely

(Bracketing) ∞ 0 log N [] , G, || • || Q,2 d < ∞,
for some particular probability measure Q, (see, e.g., [92, Definition 2.1.6] for more details). For example, a class G with square integrable envelope is Q-Donsker under this assumption. As shown in the preceding subsection, the local empirical process T n (•, h, z) can be more or less considered as an empirical process with underlying sample (Y toward Q for a strong enough seminorm is the key argument to prove that all the results of §2.1.3 are still true when (U nif. entropy) is replaced by (Bracketing), with the choice of Q as the uniform distribution on [-M, M ] d . Indeed, the reader probably noticed, throughout the reading of §2.1.3, that assumption (U nif. entropy) was used at only one single step : when bounding the expectation of suprema of empirical processes over classes of functions fulfilling (U nif. entropy) that are contained in small || • || λ,2 -balls (see Lemma 2.1.2). Using, e.g., [START_REF] Van Der | Asymptotic Statistics[END_REF]Lemma 19.34], it is possible to obtain similar bounds for classes of functions fulfilling (Bracketing) that are contained in small || • || λ,2 brackets. I hope that those few words can convince the reader that all the results of the present chapter and of Chapter 3 implying the local empirical processes T n (•, h, z) can be extended, when replacing the uniform entropy number assumptions by their bracketing counterparts, with Q being the uniform distribution on [-M, M ] d .

(h,z) i ) i≥1 having distribution

Perspectives

My main perspective of improvement of these ULDP concerns the uniform boundedness of G (assumption (Bounded)). The boundedness of G plays its crucial role in the use of Talagrand's concentration inequality, so as the second term exp(-A 2 t/M ) become negligible in front of the first (see (2.28)). Following an idea already present in [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF], I would like to relax the boundedness assumption to a (high order) integrability of the envelope, perhaps at the price of truncating, by making use of Fuk-Nagaev type inequalities. Einmahl and Li [START_REF] Einmahl | Characterization of lil behavior in banach space[END_REF] did improve the Fuk-Nagaev inequality which, in its improved form, and applied to empirical processes, can be stated as follows.

Fact 2.1.3 (Fuk-Nagaev type inequality, Einmahl, Li, 2008) Take the notations of Fact 2.1.2 but, instead of assuming (Bounded), suppose that the class G admits an envelope G which satisfies E G p (Z 1 ) < ∞ for some p > 2. Then given η ∈ (0, 1] and δ > 0, there exists C such that, for all t > 0 and n ≥ 1, we have :

P max m=1,...,n sup g∈G m i=1 g(Z i ) -E g(Z i ) ≥ (1 + η)β n + t ≤ exp - t 2 (2 + δ)nσ 2 G + CE G p (Z 1 ) n t p .

CHAPTER 2. UNIFORM LARGE DEVIATIONS AND POISSONIZATION TOOLS FOR LOCAL EMPIRICAL PROCESSES

Poissonization

Consider a sequence (η n ) n≥1 of random variables on (Ω, A, P). The η n -randomized local empirical process is defined as follows, with the convention

∅ = 0. Π ηn (g, h, z) := ηn i=1 g h,z Z i -E g h,z Z i ,
g 1 → ηn i=1 g 1 (Z i ), . . . , g p → ηn i=1 g p (Z i )
is mutually independent, as processes defined respectively on G 1 , . . . , G p .

Poissonization can be seen a way to switch from a probability calculus on empirical processes to a related calculus for their Poissonized versions, in order to use the preceding property.

The negligible effect of sample size randomization for large deviation events

Up to far in the literature, Poissonization techniques heavily relied on some interplay between Poisson and binomial distributions (see, e.g., [START_REF] Shorack | Empirical Processes and applications to statistics[END_REF]Chapter 8], [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF]Chapter 3.5]). My first main contribution to these techniques is that, when considering large deviation events, neither the fact that η n is Poisson nor that it is independent of the sample are required to obtain suitable good approximations. It is sufficient that the η n concentrate enough around their expectations n, in regard with the large deviation principle that is involved (hence the "Poissonization" terminology should be relaxed to a general sample size randomization). Our next proposition explicitly uses the ULDP framework of Proposition 2.1.2, to make the arguments more clear an coherent. However, it has to be noted that such techniques are not limited to local empirical processes. They can be translated into results of the same type for general empirical processes. Proposition 2.2.2 (Varron, 2014, from [START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF]) Take all the notations and assumptions of 2.2. POISSONIZATION Proposition 2.1.2. Assume that the sequence η ñk fulfills the following condition :

∀δ > 0, lim k→∞ sup h∈[ hk , hk ] 1 v k (h) log P η ñk ñk / ∈ [1 -δ, 1 + δ] = -∞, (2.34)
then we have, for each > 0 :

lim k→∞ sup z∈H, ||z -z|| d ≤ k h∈[ hk , hk ] 1 v k (h) log P Π η ñk ñk (•, h, z ) -T ñk (•, h, z ) G 2f (z)ñ k h d v k (h) > = -∞.
As a consequence, Proposition 2.1.2 is still true with the formal replacement of

T ñk (•, h, z ) by Π η ñk (•, h, z ).
Before giving a short proof, I would like to point out that, at least in the theory of functional limit laws, assumption (2.34) is always very largely satisfied when needed.

Proof : First fix B > 0, and set δ

:= C 2 2 /(8B∆ 2 G C 2 1 ), where C 1 and C 2 are as in Fact 2.1.2. Writing N k (δ) := n ∈ N, n/ñ k ∈ [1 -δ, 1 + δ] , we have, for fixed k ≥ 1, z ∈ H, h ∈ [ hk , hk ], || z -z ||≤ k : P Π η ñk ñk (•, h, z ) -T ñk (•, h, z ) G 2f (z)ñ k h d v k (h) > ≤P η ñk / ∈ N k (δ) + P max n∈N k (δ) sup g∈G ñk i=n+1 g h,z (Z i ) -E g h,z (Z i ) 2f (z)ñ k h d v k (h) > ,
with the convention

q i=p+1 u i := - p i=q+1
u i for p > q. By (2.34) it is sufficient to bound the second term. This is done by applying Proposition 2.1.3, noticing that, for all large enough k, we have :

sup h∈[ hk , hk ] h = hk ≤ h 0 , k ≤ 0 , u k (h) := 2 4δC 2 1 v k (h) ≥ (C 0 C 3 ) 2 2 d+1 M d , for all h ∈ [ hk , hk ], inf h∈[ hk , hk ] [δñ k + 1]h d u k = [δñ k + 1] hd k u k ≥ K 0 , which yields, ñk ≥ δ -1 , and uniformly in h ∈ [ hk , hk ], || z -z ||≤ k : P max n∈N k (δ) sup g∈G ñk i=n+1 g h,z (Z i ) -E g h,z (Z i ) 2f (z)ñ k h d v k (h) > ≤2P max 0≤m≤[δñ k +1] sup g∈G m i=1 g h,z (Z i ) -E g h,z (Z i ) 2f (z)ñ k h d v k (h) > /2 ≤2P max 0≤m≤[δñ k +1] sup g∈G m i=1 g h,z (Z i ) -E g h,z (Z i ) ≥ C 1 f (z)[δñ k + 1]h d 2 4δC 2 1 v k (h) ≤8 exp - C 2 2 v k (h) 4δC 2 1 ∆ 2 G ≤8 exp -2Bv k (h) ,
which concludes the proof, as B was arbitrarily chosen.

A general Poissonization tool for i.i.d. sums in a measurable semigroup

In this subsection I will expose a Poissonization technique which heavily relies on the interplay between binomial and Poisson distributions. Einmahl [START_REF] Einmahl | Multivariate empirical processes[END_REF] established a Poissonization technique for the local uniform empirical process. In the setup of general empirical processes, his result can be stated as follows. Recall the general notation

Q(f ) = f dQ. Proposition 2.2.3 (Derived from J. Einmahl, 1986) Let J ∈ X such that P Z 1 ∈ J < 1.
Let G be a class of functions such that g(z) = 0 for each z / ∈ J and g ∈ G. There exist a constant C, depending only upon P Z 1 ∈ J , such that, for any A ⊂ R G , which is measurable for each G n (•), n ≥ 1, we have :

∀n ≥ 1, P n i=1 δ Z i (•) ∈ A ≤ CP ηn i=1 δ Z i (•) ∈ A ,
where η n is Poisson with expectation n, and is independent of the Z i in the RHS. Moreover, when P Z 1 ∈ J ≤ 1/2, C can be chosen equal to 2.

That result turned out to be crucial when proving global functional limit laws for the local empirical process. Indeed, when G satisfies (Support), and a soon as P Z 1 ∈ H M h 0 < 1 for some h 0 > 0 then all the function g h,z , z ∈ H, h ∈]0, h 0 [, g ∈ G have their supports included in J := H M h 0 . Hence, Proposition 2.2.3 allows to Poissonize probabilities for events simultaneously involving local empirical processes at different points z and at different bandwidths h.

Giné et. al. [START_REF] Giné | The L 1 -norm density estimator process[END_REF] put Einmahl's result in the more abstract framework of sums of i.i.d. random variables (X i ) i≥1 in a measurable semigroup (D, +, D). Proposition 2.2.4 (Giné, Mason, Zaitsev, 2001) Let (X i ) i≥1 be i.i.d random variables in a measurable semigroup (D, +, D). Let B ∈ D such that P X 1 ∈ B < 1. Then there exists a constant C such that, for each measurable map H from (D, D) to [0, ∞[, we have :

∀n ≥ 1, E H n i=1 1 B (X i )X i ≤ CE H ηn i=1 1 B (X i )X i ,
where η n is Poisson with expectation n, and is independent of the X i in the RHS. Moreover, when P X 1 ∈ B ≤ 1/2, the constant C can be chosen equal to 2.

When working on spatial functional limit laws [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF], I was seeking a way to Poissonize maximal inequalities for sums of random variables in a Banach space, namely, to obtain inequalities of the form We shall write, for simplicity of notations,

∀n ≥ 1, P max m=1,...,n m i=1 1 B (X i )X i ≥ ≤ CP max m=1,...,ηn m i=1 1 B (X i )X i ≥ .
→p i=q d i := d q , d q + d q+1 , . . . , p i=q d i , when p ≥ q, :=0 otherwise.
Hence, the symbol →p i=1 is understood as the history record of partials sums up to p.

My Poissonization result is as follows :

Proposition 2.2.5 (Varron, 2011, from [START_REF] Varron | A note on weak convergence, large deviations, and the bounded approximation property[END_REF]) Let (X i ) i≥1 be i.i.d random variables in a measurable semigroup (D, +, D). Let D := n≥1 D ⊗n . Let (E 0 , A 0 ) be a measurable space and let φ : ( D, D) → [0, . . . , ∞[ be measurable. Let B ∈ D such that P X 1 ∈ B < 1. Then there exists a constant C, depending only upon P X 1 ∈ B , such that, :

∀n ≥ 1, E φ →n i=1 1 B (X i )X i ≤ CE φ →ηn i=1 1 B (X i )X i ,
where η n is Poisson with expectation n, and is independent of the X i in the RHS.

sup n≥1 n(1-p B ) [n(1-p B )] e n(1-p B ) [n(1-p B )]! n n e n n! =: C < ∞, (2.35)
and direct calculation for small values of n shows that C ≤ 2 when p B ≤ 2 (see [46, p. 8]). Now denote by (τ i , Y i ) i≥1 an i.i.d. sequence for which Y i is independent of τ i , P(τ i = 1) = 1 -P(τ i = 0) = p B and Y i has the distribution of X i conditionally to

X i ∈ B. A simple calculation shows that 1 B (X i )X i i≥1 = d τ i Y i i≥1
, from where, by conditioning on (τ 1 , . . . , τ n ) :

E φ →n i=1 1 B (X i )X i = P⊂{1,...,n} p P B (1 -p B ) n-P E φ →n i=1 1 P (i)Y i ,
with the notation P for the number of elements of P. Now notice that, for fixed P and for each permutation of indices σ, we have

1 P (1)Y 1 , . . . , 1 P (n)Y n = d 1 P (σ(1))Y 1 , . . . , 1 P (σ(n))Y n .
By choosing σ such that (1 P (σ(1)), . . . , 1 P (σ(n))) = (1, . . . , 1, 0, . . . , 0) the vector

→n i=1 1 P (σ(i))Y i
has his last n -P + 1 coordinates equal, from where, since φ is truncating :

E φ →n i=1 1 B (X i )X i = n k=0 n k p k B (1 -p B ) n-k E φ →k i=1 Y i .
The remainder of the calculus continues as in the proof of Lemma 2.1 in [START_REF] Giné | The L 1 -norm density estimator process[END_REF], namely, introducing independent Poisson random variables η B and η B C with respective expectations np B and n(1 -p B ), which are also independent of (Y i ) i≥1 :

n k=0 n k p k B (1 -p B ) n-k E φ →k i=1 Y i = n!e n n n n k=0 (np B ) k k!e k n(1 -p B ) n-k (n -k)!e n-k E φ →k i=1 Y i = 1 P η n = n n k=0 E φ →k i=1 Y i 1 {η B =k} P η B C = n -k ≤ max k=0,...,n P η B C = n -k P η n = n E φ →η B i=1 Y i = P η B C = [n(1 -p B )] P η n = n E φ →η B i=1 Y i ≤CE φ →η B i=1 Y i , by 2.35.
The end of the proof now requires an additional work, which consists in proving that

E φ →η i=1 τ i Y i = E φ →η B i=1 Y i ,
This is done by using the interplay between Poisson and Binomial distributions :

E φ →η i=1 τ i Y i = m≥0 P(η = m) P⊂{1,...,m} p P B (1 -p B ) m-P E φ →m i=1 1 P (i)Y i = m≥0 n m m! e -n m k=0 m k p k B (1 -p B ) m-k E φ →k i=1 Y i
(by the same truncating arguments as above)

= k≥0 E φ →k i=1 Y i m≥k n m m! m! k!(n -k)! p k B (1 -p B ) m-k e -n = k≥0 E φ →k i=1 Y i e -np B (np B ) k k! m ≥0 (n(1 -p B )) m m ! e -n(1-p B ) = k≥0 P(η B = k)E φ →k i=1 Y i ,
which proves Proposition 2.2.5.

Perspectives

As a perspective for future works, I would like to investigate the possible applications of Proposition 2.2.5. Since a wide variety of maps do satisfy the truncating property, such a Poissonization tool may open new research directions for empirical processes, or more generally, to i.i.d. sums in a semigroup. Some leads of applications are stated in [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF].

Chapter 3

Contributions to the study of local empirical processes

My contributions to the study of local empirical processes can be decomposed into three parts :

-The independence between bandwidths phenomenon, ands its consequences to in-bandwidth uniformity in the standard functional limit laws. Those results are stated in §3.2. -Spatial-type limit results for the local empirical process : endowing a set H with a finite measure µ, I studied the asymptotic repartitions of the T n (•, h n , z), when z is distributed by µ. Those results are stated in §3.3. -A collection of more or less incremental improvements in the existing functional limit laws, without considering in-bandwidth uniformity or spatial repartitions. Those results are stated in the following section. Each of the presented results is accompanied with some words about the key elements of its proof, with frequent references to Chapter 2.

3.1 Improvements of existing functional limit laws 3.1.1 Some preliminary words on the usual tools used in functional limit laws

In the proof of functional limit laws, several arguments are recurrent. In this subsection, I will briefly explain them, in order to mention them when needed in the sequel.

Blocking arguments

In both local and global functional limit laws, the main idea is to prove that probabilities of events A n , involving collections of processes T n (•, h, z), are a summable in n, in order to use the Borel-Cantelli lemma. Systematically, it appears that those probabilities are summable only along subsequences of the form n k := [(1 + γ) k ], for γ > 0. A key argument to interpolate between the n k is to use maximal inequalities for partial sums of i.i.d. processes. One of them is the Montgomery-Smith inequality, which I chose to write down here, for its concision and simplicity.

Fact 3.1.1 (Montgomery-Smith, 1993) Let (E, || • ||) be a Banach space, and let (X i ) i≥1 be an i.i.d. sequence, for which || X 1 || is Borel measurable. Then for all n ≥ 1 and for all t ≥ 0, we have :

P max m=1,...,n m i=1 X i ≥ t ≤ 9P n i=1 X i ≥ t 30 .
Interpolating between the n k is (up to small technicalities due to the variation of h n with n) is essentially finding bounds, for fixed > 0, for probabilities

P max m=1,...,n k -n k-1 m i=1 g h,z (Z i ) -E g h,z (Z i ) ≥ 2f (z)n k h d n k log log(n k ) ≤9P T n k -n k-1 (•, h, z) G ≥ 30 2f (z)n k h d n k log log(n k ) , ≤9P T n k -n k-1 (•, h, z) G ≥ 30 √ γ 2f (z)h d n k (n k -n k-1 ) log log(n k ) ,
where, in the last inequality, the term √ γ could be inserted because n k -n k-1 ≤ γn k for all large k. Next, a use, e.g. of Proposition 2.1.3, provides the right deviation bounds, with a choice of γ small enough to make the threshold / √ γ large enough.

Stochastic renewal

Local functional limit laws use the converse part of the Borel-Cantelli lemma. Such a converse part only involves mutually independent events A n . The T n (•, •, •) clearly fail to be independent in n. However, along sufficiently spread subsequences n k , these processes can be proved to be "almost" independent. By sufficiently spread, we can typically choose sequences for which n k-1 /n k tends to 0 (or at least, has a negligible upper limit). The idea is to first prove asymptotic results for the mutually independent sequences of processes

T k (g, h, z) := n k i=n k-1 +1 g h,z (Z i ) -E g h,z (Z i ) 2f (z)n k h d log log(n k ) , k ≥ 1 (3.1)
for which the normalization makes senses since n k -n k-1 ∼ n k . Then the distance to the original sequence can be rendered negligible by the following calculus

T k (•, h, z) - T n k (•, h, z) 2f (z)n k h d log log(n k ) = n k-1 n k × T n k-1 (•, h, z) 2f (z)n k-1 h d log log(n k ) ,
where the second factor is properly normalized to be stochastically controlled, and the first factor (n k-1 /n k ) 1/2 tends to zero.

Perspectives

A careful reading of the proof of Fact 3.1.1 (see [START_REF] Montgomery-Smith | Comparison of sums of identically distributed random vectors[END_REF]) shows that neither the Banach structure, nor the vector space structure does play a role. A perspective of future works wound be to extend Fact 3.

C 1 | k | α 1 || e ||≤|| k e ||≤ C 2 | k | α 2 || e ||,
where neither C 1 , C 2 nor α 1 , α 2 depends on k ∈ Z or e ∈ D.

Clustering rates and Chung-Mogulskii functional limit laws for the local uniform empirical process

When d = 1, G is the particular class

G 0 := 1 [0,t] , t ∈ [0, 1] ,
and when the Z i are uniform on [0, 1], the T n (•, h, z) are identified to the functional increments of the uniform empirical process. Those processes were the first to be studied (see, e.g., [START_REF] Shorack | Empirical Processes and applications to statistics[END_REF]), because, in addition to being rich enough to have connections with a lot of statistical procedures, they benefit of particularly strong mathematical tools. I will expose some of them in the next paragraph. It first seems convenient to introduce notations that are adapted to the specific framework of the uniform empirical process. We shall write the empirical process

α n (t) := √ n(F n (t) -t),
where

F n (t) := 1 n n i=1 1 [0,t] (U i ), t ∈ [0, 1],
and the (U i ) n≥1 are independent, identically distributed (i.i.d) random variables on (Ω, A, P), uniformly distributed on [0, 1]. Define the quantile process by

β n (t) = √ n F -1 n (t) -t , t ∈ [0, 1], where F -1 n (t) := inf{u : F n (u) ≥ t}.
We shall denote by W a Wiener Process on [0, 1], and by B a Brownian bridge on [0, 1] (the law of the Brownian bridge is, e.g., defined as the law of W (•) -W (1)• on [0, 1]). Throughout this subsection we will place ourselves in the metric space

( ∞ ([0, 1]), || • || [0,1]
).

First tool : Strong approximation by Gaussian processes

Strong approximation of empirical processes by their Gaussian counterparts has been the object of several research works [START_REF] Einmahl | Uniform in bandwidth consistency of variable bandwidth kernel estimators[END_REF][START_REF] Koltchinskii | komlós-major-tusnády approximation for the general empirical process and haar expansions of classes of functions[END_REF][START_REF] Komlós | An approximation of partial sums of independent r.v.'s and the sample d.f.I[END_REF][START_REF] Komlós | An approximation of partial sums of independent r.v.'s and the sample d.f.II[END_REF][START_REF] Massart | Strong approximation for multivariate empirical and related processes, via kmt constructions[END_REF][START_REF] Révész | On strong approximation of the multidimensional empirical process[END_REF][START_REF] Rio | Vitesses de convergence dans le principe d'invariance faible pour la fonction de répartition empirique multivariée[END_REF][START_REF] Tusnady | A remark on the approximation of the sample df in the multidimensional case[END_REF]. Among them, the Komlos-Major-Tusnady construction [START_REF] Komlós | An approximation of partial sums of independent r.v.'s and the sample d.f.I[END_REF][START_REF] Komlós | An approximation of partial sums of independent r.v.'s and the sample d.f.II[END_REF][START_REF] Tusnady | A remark on the approximation of the sample df in the multidimensional case[END_REF] for the uniform empirical process is the only one that provides rates of approximations of order log(n)/ √ n. These rates of approximations have been proved to hold for the bivariate uniform empirical process [START_REF] Komlós | An approximation of partial sums of independent r.v.'s and the sample d.f.II[END_REF] (replacing log(n) by log(n) 2 in the rate of approximation). Fact 3.1.2 (Komlos-Major-Tusnady [START_REF] Komlós | An approximation of partial sums of independent r.v.'s and the sample d.f.I[END_REF]) At the price of enriching the probability space (Ω, A, P), one can construct a sequence B n of copies of B, such that, for some constants C 1 , C 2 , C 3 we have, for all n ≥ 1 and z > 0 :

P || α n -B n || [0,1] ≥ C 1 log n + z √ n ≤ C 2 exp -C 3 z .
A similar result was proved by the authors [START_REF] Komlós | An approximation of partial sums of independent r.v.'s and the sample d.f.II[END_REF] for the standard Poisson process on the real line, involving copies W n of W . Such a powerful strong approximation result not only allows to prove appropriate large deviation principles, but it also allows to explore second order convergence in these large deviations. For example, while a large deviation result may only give rates of convergence for open ball with fixed radius (namely O = ψ in Proposition 2.1.2), the KMT approximation allows, by Gaussian analysis, to investigate balls with radii tending to 0 as n → ∞ (namely O = ψ n , with n → 0).

Second tool : Gaussian analysis for enlarged Strassen sets

The description of the first tool leads to the question of small ball probabilities (or more generally, probabilities involving A n , for a fixed region A). When d = 1, Gaussian analysis provides sharp bounds for these probabilities. When d ≥ 2, generalizing such results is an open an tough problem (see, e.g., [START_REF] Lifshits | Lectures on Gaussian processes[END_REF] for an overview on this field). Define the Strassen sets

S 1 := f ∈ ∞ ([0, 1]), ∃f Borel, f (•) := • 0 f (t)dt, 1 0 f 2 (t)dt ≤ 1 , and
S 2 := f (t) ∈ S 1 , f (1) = 0 .
Note that S 2 (resp. S 1 ) is the unit ball of the reproducing kernel Hilbert space of the Brownian bridge (resp. of the Wiener process) on [0,1]. The next inequalities, due to Talagrand, involve small enlargements of those Strassen sets (recall (2.5)).

IMPROVEMENTS OF EXISTING FUNCTIONAL LIMIT LAWS

Fact 3.1.3 (Talagrand [START_REF] Talagrand | New Gaussian estimates for enlarged balls[END_REF]) There exists two constants L 0 and u 0 >0 such that, for any 0 < u < u 0 and c > 0, we have :

P B / ∈ (cS 2 ) u ≤ exp L 0 u 2 - cu 2 - c 2 2 .
There exist two constants u 1 and L 1 such that, for any 0 < u < u 1 and c > 0, we have

P W / ∈ (cS 1 ) u ≤ exp L 1 u 2 - cu 2 - c 2 2 .
Berthet [START_REF] Berthet | On the rate of clustering to the Strassen set for increments of the empirical process[END_REF] used inequality (3.2) coupled with the KMT approximation of Fact 3.1.2 to obtain clustering rates in the global standard functional limit laws for the functional increments of the uniform empirical process. In the same spirit, I proved a similar result for the local standard law of Mason [START_REF] Mason | A strong invariance principle for the tail empirical process[END_REF] and for the functional limit law of Finkelstein [START_REF] Finkelstein | The law of iterated logarithm for empirical distributions[END_REF]. Those are stated in the next two theorems.

Theorem 2 (Varron, 2011 [102]) There exists a universal constant 0 >0 such that, for any choice of > 0 we have almost surely, for all large n :

α n 2 log log(n) ∈ S (log log(n)) -2/3 2 , β n 2 log log(n) ∈ S (log log(n)) -2/3 2 .
Theorem 3 (Varron, 2011 [102]) Let h n be positive real numbers satisfying, as n → ∞,

nh n ↑ ∞, nh n (log log(n)) 7/3 → ∞, h n ↓ 0. (3.2)
Then there exists a universal constant 1 > 0 such that, for any choice of > 1 we have almost surely, for all large n :

α n (h n •) 2h n log log(n) ∈ S (log log(n)) -2/3 1 .
If, in addition, nh n /(log log(n)) 11/3 → ∞, then we have, almost surely, for all larg n :

β n (h n •) 2h n log log(n) ∈ S (log log(n)) -2/3 1 .

Third tool : Small ball probabilities

The two preceding theorems clearly establish outer clustering rates in Strassen functional limit laws. Another interesting topic is to establish inner rates of approximations of fixed elements of S 1 (or S 2 ). Such results are known under the name of functional Chung-type limit laws. Those inner rates involve small ball probabilities, which have been intensively investigated for the Wiener process W . With the help of the works of de Acosta [START_REF] Acosta (de | Small deviations in the functional central limit theorem with applications to the functional laws of he iterated logarithm[END_REF], Deheuvels [START_REF] Deheuvels | Chung type functional laws of the iterated logarithm for tail empirical processes[END_REF] established a Chung-type limit law for (α n (h n •)) n≥1 , by showing that, if h n is a sequence of constants satisfying nh n ↑ ∞, h n ↓ 0 and nh n /(log log(n)) 3 → ∞, we have, almost surely, for each f ∈ S 1 satisfying || f || 2

Hilb :=

1 0 f 2 (t)dt < 1 : lim n→∞ (log log(n)) α n (h n •) √ h n b n -f [0,1] = π 4 1-|| f || 2 Hilb .

His result, however, left open the problem of finding inner rates when || f || 2

Hilb = 1. Since the works of Acosta, several researchers managed to obtain small ball probability estimates for elements f pertaining to the border of S 1 . Such estimates were obtained for two particular classes of functions, which are defined as follows. the subset of S 1 of functions f for which 1 0 f 2 (t)dt = 1, and for which f admits a version which is locally of infinite variation, namely : there exist x 1 , . . . , x k ∈ [0, 1] such that, for > 0 we have (V, denoting the total variation on [0, 1])

V k i=1 f 1 {[0,1]-[x i -,x i + ]} < V k i=1 f 1 {[x i -,x i + ]} = ∞.
The works of Csáki [START_REF] Csàki | A relation between Chung's and Strassen's laws of the iterated logarithm[END_REF], Grill [START_REF] Grill | A lim inf result in Strassen's law of the iterated logarithm[END_REF], Gorn and Lifshits [START_REF] Gorn | Chung's law and the Csàki function[END_REF], and Berthet and Lifshits [START_REF] Berthet | Some exact rates in the functional law of the iterated logarithm[END_REF] on small ball probabilities for Wiener processes can be summed up in a single inequality. Inequality 3.1.1 (From Csáki, Grill,Gorn, Berthet, Lifshits [START_REF] Csàki | A relation between Chung's and Strassen's laws of the iterated logarithm[END_REF][START_REF] Grill | A lim inf result in Strassen's law of the iterated logarithm[END_REF][START_REF] Gorn | Chung's law and the Csàki function[END_REF][START_REF] Berthet | Some exact rates in the functional law of the iterated logarithm[END_REF]) For any f ∈ S BV 1 ∪S LIV

1

, there exists a function ∇ f (•) from ]0, ∞[ to itself, as well as a constant χ f > 0, such that the following assertion holds : for each δ > 0, there exist γ + = γ + (δ, f ) > 0 and γ-= γ -(δ, f ) > 0 such that for all T sufficiently large :

P ∇ f T 2 2 W T -f [0,1] ≤ (1 + δ)χ f ≥ exp - T 2 2 + γ + ∇ 2 f (T 2 /2) T 2 , P ∇ f T 2 2 W T -f [0,1] ≤ (1 -δ)χ f ≤ exp - T 2 2 -γ -∇ 2 f (T 2 /2) T 2 .
Note that ∇ f and χ f do have explicit forms. However, for sake of simplicity, I chose not to give the details of their definitions here. My contribution to Chung-Mogulskii functional limit laws is as follows. It is the expected counterpart of Berthet's result in the global standard functional limit laws [START_REF] Berthet | Inner rates of coverage of Strassen type sets by increments of the uniform empirical and quantile processes[END_REF].

IMPROVEMENTS OF EXISTING FUNCTIONAL LIMIT LAWS

Theorem 4 (Varron, 2011 [102]) Let f ∈ S bv 1 ∪ S liv 1 be arbitrary and let h n be a sequence of real numbers satisfying, as n → ∞,

nh n ↑ ∞, h n ↓ 0, lim n→∞ nh n log log(n)∇ 2 f (log log(n)) = ∞. (3.3)
Then we have, almost surely :

lim n→∞ ∇ f (log log(n)) α n (h n •) 2h n log log(n) -f [0,1] = χ f .

Some words about the proofs

In each theorem, the growth assumption made upon nh n is slightly stronger than the usual condition nh n / log log(n) → ∞. The sequence log log(n) is replaced by a power of log log(n), similarly as in [START_REF] Berthet | On the rate of clustering to the Strassen set for increments of the empirical process[END_REF][START_REF] Berthet | Inner rates of coverage of Strassen type sets by increments of the uniform empirical and quantile processes[END_REF]. We imposed that condition so that the KMT strong approximation can be sharp enough to be negligible in front of the small ball/ small enlargements Gaussian analysis. Beside the crucial use of the KMT approximation and the above-mentioned Gaussian inequalities, the proof is routine use of blocking and stochastic renewal arguments, with contextual minor improvements in regard of the existing literature. Note that, in the original article [START_REF] Varron | Clustering rates and chung type functional laws of the iterated logarithm for empirical and quantile processes[END_REF], I needed to impose an additional assumption on h n , by requiring that this sequence is not too slow, namely h n log log(n) → 0.

(3.4)

This assumption was imposed in order to use a Poissonization technique ([102, Lemma 3.2]), which was later made obsolete by the Poissonization written in Proposition 2.2.2. That later Proposition could then get rid of (3.4). To conclude this paragraph, the results involving β n and β n (h n •) are obtained by the Bahadur-Kiefer representation [START_REF] Kiefer | On Bahadur's representation of sample quantiles[END_REF] and its local version [39, Theorem 5].

Perspectives

Berthet [START_REF] Berthet | Functional chung laws for small increments of the empirical process and a lower bound in the strong invariance principle[END_REF] showed that, for the problem of small balls around functions f ∈ S BV 1 , the KMT approximation could be bypassed by making use of estimates of small ball probabilities for Poisson processes with high intensity, due to Shmileva [START_REF] Yu | Small ball probabilities for a centered poisson process of high intensity[END_REF]. Using those estimates, and applying them in the context of the global standard functional limit law for the increments of α n , Berthet [START_REF] Berthet | Functional chung laws for small increments of the empirical process and a lower bound in the strong invariance principle[END_REF] could exhibit an uncrossable lower bound for the strong approximation of α n by Brownian bridges, namely : Fact 3.1.4 (Berthet, 2010 [10]) Let (Ω, A, P) be a probability space, let (U i ) i≥1 be a sequence of i.i.d. random variable uniformly distributed on [0, 1], let (α n ) n≥1 be the corresponding sequence of uniform empirical processes, and let B n be a sequence of Brownian bridges on (Ω, A, P). Then we have, almost surely :

lim n→∞ √ n log(n) || α n -B n || [0,1] ≥ 1 6 .
That lower bound is explained by the fact that α n does have an intrinsic dissymmetry at small scales, which makes him approximate more closely the target functions f which satisfy f (t) 3 dt > 0 than a Brownian bridge does. That dissymmetry comes from the fact that α n is the difference between the stepwise increasing function F n and the continuously decreasing identity function on [0, 1]. For the local empirical process α n (h n •) Mason and van Zwet [START_REF] Mason | A refinement of the KMT inequality for the uniform empirical process[END_REF] did prove a strong approximation result which takes in account the fact that h n → 0, and hence is more precise that using B n (h n •) in the KMT approximation (recall Fact 3.1.2). An interesting perspective is to investigate how the estimates of Shmileva [START_REF] Yu | Small ball probabilities for a centered poisson process of high intensity[END_REF] could give information on an uncrossable lower bound for the strong approximations of α(h n •).

Nonstandard functional limit laws

Let us close this parenthesis on the uniform empirical process, and consider again the general objects T n (•, h, z) In this subsection, the centering in not necessary, so we will rather focus on the process

T n (g, h, z) := n i=1 g h,z (Z i )
When H = {z 0 } is a singleton, and when considering a bandwidths sequence h n fulfilling nh d n ∼ c log log(n) for c ∈]0, ∞[, the asymptotic behavior of T n (•, h n , z 0 ) is not Gaussian anymore. Indeed, the inherent large deviation principle and limit set are related to large deviations of Poisson random measures.

I G (ψ) := inf [-M,M ] d h 1 g dλ, ∀g ∈ G, ψ(g) = [-M,M ] d ggdλ ,
where h 1 (x) := x log(x)-x+1 for x ≥ 0 (with an extension by continuity at 0), and h 1 (x) = ∞ otherwise. Define the associated levels

Γ G,a := ψ ∈ ∞ (G), I G (ψ) ≤ a = g → [-M,M ] d ggdλ, [-M,M ] d h 1 g dλ ≤ a .
The first (local) nonstandard functional limit law was stated by Deheuvels and Mason [START_REF] Deheuvels | Nonstandard functional laws of the iterated logarithm for tail empirical and quantile processes[END_REF], for the uniform empirical process. It was then extended by Deheuvels and Mason to a multivariate setting, with G being a class of indicators of sets which can be included in finite reunions of || • || λ,2 brackets generated by finite partitions of a set [-M, M ] d . (see, [START_REF] Deheuvels | Nonstandard local empirical processes indexed by sets[END_REF] for more details). Their two basic ingredients are a strong approximation by a sequence of Poissonized local empirical processes Π n (•, h n , z 0 ), (which is possible since h n tends rapidly to 0), then the use of the fact that, on an arbitrary partition (A 1 , . . . , A p ) of [-M, M ] d a Poisson random measure defines a mutually independent family. They then use that independence to tensorize the large deviation properties of each Π n (1 A j , h n , z 0 ) and hence obtain a large deviation principle for the joint laws Π n (1 A j , h n , z 0 ) j≤p . With M. Maumy-Bertrand [START_REF] Maumy | Non standard functional limit laws for the increments of the compound empirical distribution function[END_REF], we provided an extension of their result to processes of the type T n,c (g, •, h, z), where

T n,c (g, g, h, z) := n i=1 g h,z (Z i )g(Y i ),
for a function g taking values in R k , for which we made assumptions of finite conditional exponential moments for g(Y 1 ). Such a nonstandard functional limit law also occurs when H is a compact with nonempty interior , as showed by Deheuvels and Mason [START_REF] Deheuvels | Functional laws of the iterated logarithm for the increments of empirical and quantile processes[END_REF] for the uniform empirical process (in which case it is required that nh d n ∼ c log(n) instead of c log log(n)). My contribution (in [START_REF] Varron | A nonstandard uniform functional limit law for the increments of the multivariate empirical distribution function[END_REF]) is a generalization of the set-indexed local law of Deheuvels and Mason [START_REF] Deheuvels | Nonstandard local empirical processes indexed by sets[END_REF] to a global law. In its published form, it is restricted to the class

G 0,d := 1 [0,t 1 ]×...×[0,t d ] (•), (t 1 , . . . , t d ) ∈ [0, 1] d .
Theorem 5 (Varron, 2010, [100]) Assume that the law of Z 1 fulfills (Hf ) for some open set O. Let H ⊂ O be a compact set with nonempty interior. Let h n be a bandwidths sequence such that nh d n ∼ c log(n) for some c ∈]0, ∞[. Then, almost surely, the two following assertions hold Proof of the corollary : Identifying the T n (•, h n , z) to random distribution functions on [0, 1] d , and by integration by parts, the preceding theorem can be directly generalized to arbitrary finite classes G of functions on [0, 1] d that are of bounded variation (in the sense that they are themselves distribution functions of finite signed measures on [0, 1] d ). This is a consequence of the fact that, for such a function g, the map ψ →

(i) lim n→∞ sup z∈H inf ψ∈Γ G 0,d ,cf (z) T n (•, h n , z) nh d n -ψ G 0,d = 0, (ii)∀z ∈ H, ∀ψ ∈ Γ G 0,d ,cf (z) , lim n→∞ inf z∈H T n (•, h n , z) nh d n -ψ G 0,d = 0.
[0,1] d ψdg is continuous on ∞ ([0, 1] d ), || • || [0,1] d .
The support [0, 1] d can be extended to any [-M, M ] d with no efforts. Now, since h 1 (x) is finite only for x ≥ 0 the limit sets Γ G,a , for any G fulfilling (support) and a ≥ 0, contain elements of the form ψ : g → [-M,M ] d gddλ, where g is nonnegative. This automatically entails an ordered structure

g 1 ≺ g ≺ g 2 ⇒ ψ(g) ∈ [ψ(g 1 ), ψ(g 2 )],
where ≺ stands for the partial order on real functions. The same remark holds for the processes T n (•, h n , z). Hence, by a simple bracketing argument as in proof of the Glivenko-Cantelli theorem under bracketing (see, [92, p. 122]), Theorem 5 can be directly extended to any class of functions G satisfying (Support) and admitting finite || • || λ,2 bracketing numbers, as soon as the brackets [g, g ] are defined with g and g of bounded variation. Any bounded measurable function on [-M, M ] d can be || • || λ,2 -approximated from below and from above by functions of bounded variation, so the brackets can be taken without imposing bounded variation.

Perspective

My main perspective is to investigate wether a condition on uniform covering numbers may lead to a variant of Theorem 5. The works of Menneteau [START_REF] Menneteau | Large deviations for local empirical measures[END_REF], giving general conditions for the Poisson large deviations of local empirical measures may be a good starting point.

Uniform-in-bandwidth functional limit laws

In this section, I will give a description of my contributions to the study of the impact of the in-bandwidth uniformity to standard functional limit laws for the local empirical process. At this stage of the manuscript, it is convenient to remind the notion of Strassen-type sets. Definition 3.2.1 (Strassen-type set) Let G be a class of functions satisfying (Support) and (U nif. entropy). Recall that J G has been defined in (2.4). Given a ≥ 0, the associated Strassentype set is defined by :

S G,a := ψ ∈ ∞ (G), J G (ψ) ≤ a = g → R d ggdλ, g Borel, R d g 2 dλ ≤ a .
In the next results, we will make structural assumptions on G that may be slightly stronger that those used in the ULDP of Propositions 2.1.2 and 2.1.4. Define

G ρ := g λ -1 • -g(•), λ ∈ [1, ρ], g ∈ G , (3.5)
G +,δ := g • +u -g(•), || u || d ≤ δ, g ∈ G . (3.6)
The assumptions that will be made are (recalling (2.23)) :

(U nif. entropy dilatations) There exists ρ > 1 such that G ρ satisfies ∞ 0 sup Q probab. log N ( , G ρ , || • || Q,2 )d < ∞;
(U nif. entropy translations) There exists δ > 0 such that G +,δ satisfies

∞ 0 sup Q probab. log N ( , G +,δ , || • || Q,2 )d < ∞; (Contin. dilatations) lim ρ→1 ∆ 2 Gρ,2 = 0; (Contin. translations) lim δ→0 ∆ 2 G +,δ ,2 = 0.
Those assumptions allow to use the empirical processes theory in order to make small interpolations between finite grids of points, z ∈ H or of bandwidths h ∈ [h n , h n ]. As pointed out by Mason [65, Section 3], several classes of functions that are used in practice do satisfy those assumptions. Moreover, as mentioned in §2.1.5, all the results of the present section remain valid when the uniform entropy conditions are replaced by their bracketing counterparts.

In metric spaces, the topological limit of sequence of sets can be defined as follows.

Definition 3.2.2 (Inner and outer topological limits in a metric space) Let (E, d) be a metric space, and let (E n ) n≥1 be a sequence of subsets of E. The inner topological limit of (E n ) n≥1 is defined as When those two sets are equal, we denote them as the topological limit of (E n ) n≥1 .

Asymptotic independence in the local standard functional limit law

Proposition 2.1.4 has a direct consequence in terms of functional limit laws, when considering two local empirical processes sequences T n (•, h n , z 0 ) and T n (•, h n , z 0 ), with h n = o(h n ). Our next result is an unpublished generalization of a published result which was restricted to the local uniform empirical process. We will denote the bivariate strassen type sets as follows (recall that

J ⊗2 G (ψ 1 , ψ 2 ) := J G (ψ 1 ) + J G (ψ 2 )) S ⊗2 G,a := (ψ 1 , ψ 2 ) ∈ ∞ (G) × ∞ (G), J ⊗2 G (ψ 1 , ψ 2 ) ≤ a , a ≥ 0.
Theorem 6 (Extension of Varron, 2006 [96]) Assume that the law of Z 1 fulfills (Hf ) for an open set O. Let z 0 ∈ O. Assume that G fulfills (P ointw. sep), (Bounded), (Support), (U nif. entropy dilatations) and (Contin. dilatations). Let h n ≤ h n be two subsequences of non negative numbers such that h n = o(h n ) and

h n ↓ 0, nh d n ↑, nh d n log log(n) → ∞.
Then the sequence

T n (•, h n , z 0 ) 2f (z 0 )nh d n log log(n) , T n (•, h n , z 0 ) 2f (z 0 )nh d n log log(n) n≥2 almost surely admits S ⊗2 G,1 as cluster set in ∞ (G), || • || G × ∞ (G), || • || G .
Remark : Theorem 6 can be also stated in terms of almost sure topological limits. It states that, almost surely, the outer topological limit of the sequence of singletons

Θ local⊗2 n (h n , h n , z 0 ) := T n (•, h n , z 0 ) 2f (z 0 )nh d n log log(n) , T n (•, h n , z 0 ) 2f (z 0 )nh d n log log(n)
, is S ⊗2 G,1 , while its inner topological limit is ∅.

Some words about the proof

The essential part of the proof relies on a ULDP which is a consequence of Proposition 2.1.4, taking H = {z 0 } and k = 0. We then use blocking and stochastic renewal arguments. By looking at the definition of S ⊗2 G,1 , Theorem 6 can be roughly interpreted as follows : two local empirical processes at the same point z 0 , but at radically different scales, cannot simultaneously hit (neighborhoods of) functions both having a high stochastic cost. If the first hits a (neighborhood of) function ψ with J G close to 1, then the second has to compensate by hitting (neighborhood of) functions with low stochastic costs.

In bandwidth uniformity in the local standard functional limit law

For stating local standard functional limit laws, we will use the notations :

Θ local n (h, h, z 0 ) := T n (•, h, z 0 ) 2f (z 0 )nh d log log(n) , h ∈ [h, h] n ≥ 2, 0 < h ≤ h.
I will now state my contribution. In order to gain in concision and clarity, I will formulate it in a very slightly weaker form than that written in [START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF], in order to emphasize the notions of

UNIFORM-IN-BANDWIDTH FUNCTIONAL LIMIT LAWS

inner and outer topological limits.

Theorem 7 (Varron, 2014, from [START_REF] Varron | The almost sure Kuratowski limit of some local emprical pocesses with variable bandwidths[END_REF]) Assume that the law of Z 1 fulfills (Hf ). Assume that G fulfills (P ointw. sep.), (Bounded), (Support), (U nif. entropy dilatations), (U nif. entropy translations), (Contin. dilatations) and (Contin. translations). Let H = {z 0 } be a singleton. Let h n ≤ h n be two subsequences of non negative numbers such that, for some δ ∈ (0, 1], we have :

h n ↓ 0, nh d n ↑, nh d n log log(n) → ∞, log log(h n /h n ) log log(n) → δ. (3.7)
Then, almost surely, the inner and outer topological limits of

Θ local n (h n , h n , z 0 ) n≥2 in ∞ (G), || • || G are S G,δ and S G,1+δ .

Sketch of the proof and comments

Let us point out some key arguments of that proof.

Outer bounds

Consider, for arbitrary > 0, the closed set

F := ∞ (G) -S 2
G,1+δ , which satisfies J G (F ) > 1 + δ + 3τ for some τ > 0 (this comes from the fact that J G is a good rate function). We will focus on finding upper bounds for the probabilities P(A n ), where

A n := h∈[hn,hn] T n (•, h, z 0 ) 2f (z 0 )nh d log log(n) ∈ F .
A natural way to handle this probability of union is to discretize [h n , h n ] into a finite grid, h n, :=ρ n h n , = 0, . . . , R n , where, [•] denoting the integer part

R n := log(h n /h n ) log(ρ n ) , (3.8)
and then focus on P(A n ), where

A n := Rn =0 T n (•, h n, , z 0 ) 2f (z 0 )nh d n, log log(n) ∈ F .
This strategy is possible if we choose ρ n such that ρ n → 1. In that case the oscillations between two consecutive elements T n (•, h n, , z 0 ) and T n (•, ρ n h n, , z 0 ) are uniformly (in ) negligible, because they are suprema of empirical processes on classes G ρ , with ρ close to 1. Those are handled by using Proposition 2.1.3, which is possible because (U nif. entropy dilatations) and (Contin. dilatations) have been made. Now, using Bonferroni's inequality, we have

P(A n ) ≤(R n + 1) max =0, ...,Rn P T n (•, h n, , z 0 ) 2f (z 0 )nh d n, log log(n) ∈ F ≤(R n + 1) exp -J(F ) -τ log log(n) ,
where that last inequality holds for n large enough, by a use of Proposition 2.1.2 with ñk := n,

hk := h n , hk := h n , k = 0, v k (h) = log log(n). A choice of ρ n := exp(log(n) -τ ) entails log(ρ n ) = log(n) -τ . Now, since log log(h n /h n )/ log log(n) → δ we have R n ≤ log(n) δ log(n) τ ,
from where :

P(A n ) = O log(n) J(F )-2τ = O log(n) -1+τ .
This sequence is summable along any subsequence

n k = [(1 + γ) k ],
for arbitrarily small γ > 0. Now the interpolation between elements of that subsequence is made by making use of the usual blocking arguments. Hence, by the Borel-Cantelli lemma we have

P lim n→∞ h∈[hn,hn] T n (•, h, z 0 ) 2f (z 0 )nh d log log(n) / ∈ S 2 G,1+δ
= 0, which proves that, if it exists, the almost sure topological limit of

Θ local n (h n , h n , z 0 ) is included in S G,1+δ .

Inner bounds

It is already known ( [START_REF] Einmahl | Uniform in bandwidth consistency of variable bandwidth kernel estimators[END_REF]) that, when h n = h n , the outer topological limit of Θ n (h n , h n , z 0 ) (which, in this case, are singletons) is S G,1 . So the outer bound S G,1+δ is either too rough (because of Bonferroni's inequality), or it has a stochastic explanation. In the next lines, we will provide such an explanation. I already evoked the between bandwidths asymptotic independence phenomenon, which was illustrated in Proposition 2.1.4 and Theorem 6. Note that the just mentioned results admit a natural extension to finite collections of local empirical processes. Since

h n = o(h n ), it is possible choose ρ n → ∞ in the discretization of [h n , h n ].
This would lead to the exhibition of finite grids h n, , = 0, R n , with max ≤Rn h n, +1 /h n, → 0. Using the heuristic of independence between bandwidths, this would result in the fact that T loc n (h n , h n , z 0 ) contains R n "mutually independent" local empirical processes, with R n → ∞. Hence, even if ψ has a stochastic cost greater that 1 (J G (ψ) > 1) it is still probable enough that at least one of the R n independent processes does hit a neighborhood of ψ (because "we run many independent trials"). Write ψ for the open ball with center ψ and radius > 0 in ∞ (G). First, since J G is a rate function, we have, J G (ψ ) < 1 + δ -3τ , for some τ > 0. We now explicitly choose

UNIFORM-IN-BANDWIDTH FUNCTIONAL LIMIT LAWS

ρ n := exp log(n) τ in the discretization of [h n , h n ]. Writing B n := h∈[hn,hn] T n (•, h, z 0 ) 2f (z 0 )nh d log log(n) ∈ ψ ,
we can make the following heuristic calculus

P h∈[hn,hn] T n (•, h, z 0 ) 2f (z 0 )nh d log log(n) ∈ ψ ≥P Rn =0 T n (•, h n, , z 0 ) 2f (z 0 )nh d n, log log(n) ∈ ψ 1 - Rn =0 1 -P T n (•, h n, , z 0 ) 2f (z 0 )nh d n, log log(n) ∈ ψ by "independence" (3.9) ≥1 -exp R n min =0,...,Rn P T n (•, h n, , z 0 ) 2f (z 0 )nh d n, log log(n) ∈ ψ , since 1 -u ≤ exp(-u), u ∈ R ≥ 1 2 ∧ 1 2 R n min =0,...,Rn P T n (•, h n, , z 0 ) 2f (z 0 )nh d n, log log(n) ∈ ψ , since 1 -exp(-u) ≥ 1 2 ∧ u 2 , u ≥ 0 ≥ log(n) δ-τ log(n) -1-δ+2τ = log(n) -1+τ ,
where, again, we used the ULDP of Proposition 2.1.2 and the fact that log(n) δ-τ ≤ R n for all large n. Now by the usual renewal arguments used in Strassen laws, we can suppose that the B n are "sufficiently independent" between sufficiently spread subsequences. This is for example the case for n k ∼ exp log(k) τ /2 ), for which we also have

k≥1 P(B n k ) = ∞.
A use of the converse part of the Borel-Cantelli lemma then leads to

P lim k→∞ B n k = 1,
which can conclude the proof of the "outer topological limit" part of Theorem 7.

Explanation of the heuristic of (3.9)

We will explain how the heuristic of (3.9) is justified. As mentioned earlier, Proposition 2.1.4 admits a generalization for more than two processes. Hence, that proposition would be the appropriate tool if R n was bounded. Since it is not the case, we will approximate these local empirical processes by truly mutually independent processes. First, a use of the Poissonization technique of Proposition 2.2.2 allows to switch from

T n (•, h n, , z 0 ), ≤ R n to Π n (•, h n, , z 0 ), ≤ R n .
However, the classes of functions G := g h n, ,z 0 , g ∈ G , = 0, . . . , R n have their supports included respectively in A(h n, , z 0 ), = 0, . . . , R n (recall (2.13)). This family of sets is not mutually disjoint, so we cannot directly use Proposition 2.2.1 to achieve rigorous independence. The last trick is to construct, by successive translations of respective lengths 2M h n, , a sequence of points z n, such that, writing

| u | d := max{| u 1 |, . . . , | u d |} : z n,0 := z 0 , z n, + [-M h n, , M h n, ] d ∩ z n, + [-M h n, , M h n, ] d = ∅, if = , (3.10)
∀ ≤ R n , | z n, -z | d ≤ 2M -1 =0
h n, , from where

δ n := max =0,...,Rn h -1 n, | z n, -z 0 | d → 0. (3.11)
For example, for d = 1, we can recursively set z n, +1 := z n, + 2M h n, . Now, by (3.10), the family of processes

Π n (•, h n, , z n, ) 2f (z 0 )nh d n, log log(n) , = 0, . . . , R n
is mutually independent. Moreover, the fact that we take local processes at points different from z 0 does not hurt for using large deviations, because the ULDP of Proposition 2.1.2 allows to take z = z, a soon as || z -z || d ≤ n , for n → 0. Assertion (3.11) is largely sufficient to ensure these proximities between the z n, and z 0 (take n := δ n h n ). What remains to show is a uniform bound for the following probabilities

P Π n (•, h n, , z n, ) -Π n (•, h n, , z 0 ) G ≥ 2f (z 0 )nh d n, log log(n) , = 0, . . . , R n ,
or, equivalently, find bounds for

P T n (•, h n, , z n, ) -T n (•, h n, , z 0 ) G ≥ 2f (z 0 )nh d n, log log(n) , = 0, . . . , R n ,
in which case the Poissonization occurs after the introduction of the z n, . Now since, for any g ∈ G g h n, ,z n, (•) = g h n, ,z 0 • +h -1 n, (z n, -z 0 ) , we always have (recall (3.6)):

P T n (•, h n, , z n, ) -T n (•, h n, , z 0 ) G ≥ 2f (z 0 )nh d n, log log(n) ≤P T n (•, h n, , z 0 ) G +,δn ≥ 2f (z 0 )nh d n, log log(n) ,
Now assumptions (U nif. entropy translations) and (Contin. translations) play their only role here, to control those deviations probabilities, uniformly in ≤ R n , through the use of Proposition 2.1.3.

Asymptotic independence in the global standard functional limit law

Before entering into the subject, I will briefly revisit an important result due to Mason : the global standard functional limit law for the local empirical process.

Removing the Euclidian condition in Mason's global standard functional limit law for the local empirical process

Now consider a compact H ⊂ O with nonempty interior and consider, for a sequence of bandwidths h n , the collections of processes :

Θ global n (h n , H) := T n (•, h n ) 2f (z)nh d n log(1/h d n )
, z ∈ H .

Mason [START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF] did prove a functional limit law for Θ global n (h n , H), under the usual Csörgő-Révész-Stute conditions (see (3.12) below). His result was a major step. Indeed, in the setting of global functional limit laws, this result was the first to consider the general object T n (•, h, z), instead of the uniform empirical process, bypassing the use of the KMT approximation. Beside assumptions (P ointw. sep), (Bounded), (Support), (Contin. translations) and (Contin. dilatations), he had to strengthen the structural assumptions (U nif. entropy dilatations) and (U nif. entropy translations) to an assumption of polynomial uniform entropy numbers, namely :

(Euclidian) ∃C 0 > 0, v 0 > 0, ∀ ∈ (0, 1), sup Q probab N ( , F, || • || Q,2 ) ≤ C 0 -v 0 ,
where

F := g h,z (•), z ∈ R d , h > 0, g ∈ G .
That assumption, though more stringent that, e.g., (U nif. entropy), still encompasses Vapnik-Chervonenkis classes of sets, or more generally uniformly bounded V-C subgraph classes of functions (see, e.g. [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF]Chapter 2.6]). Mason's result is stronger than an almost sure topological limit, as it was already the case for the uniform empirical process [START_REF] Deheuvels | Functional laws of the iterated logarithm for the increments of empirical and quantile processes[END_REF]. It is a convergence for the Hausdorff distance. Note that a sequence of sets (E n ) has topological limit E when d Haus (E n , E) → 0, but the converse is not true.

Fact 3.2.1 (Mason, 2004, [65]) Assume that the law of Z 1 fulfills (Hf ) for a set O. Let H ⊂ O be a compact set with nonempty interior. Assume that G fulfills (P ointw. sep), (Bounded), (Support), (Euclidian), (Contin. translations) and (Contin. dilatations). Let h n be a sequence of bandwitdhs fulfilling the Csörgő-Révész-Stute conditions :

h n ↓ 0, nh n ↑, nh d n log(1/h d n ) → ∞, log(1/h n ) log log(n) → ∞.
Then, almost surely lim

n→∞ d Haus Θ global n (h n , H), S G,1 = 0.
In a discussion following his theorem, Mason wrote that it was "not clear at all wether assumption (Euclidian) could be relaxed to a less stringent condition" (see his remarks 1 and 2 in [START_REF] Mason | A uniform functional law of the iterated logarithm for the local empirical process[END_REF]). My contribution to the problem is as follows. I did not send it to publication.

Sketch of the proof and comments

In this subsection I will explain the technical difficulty that led Mason to strengthen (U nif. entrpoy dilatations) to (Euclidian), and how I could (somewhat simply) bypass the difficulty. Second, some aspects of his proof will come in handy to explain some heuristics of my contribution in the next paragraph.

UNIFORM-IN-BANDWIDTH FUNCTIONAL LIMIT LAWS

Outer part

To prove the outer part of Fact 3.2.1, we need to control the probabilities

P ∃z ∈ H, T n (•, h n , z) 2f (z)nh d n log(1/h d n ) / ∈ S G,1 .
The first idea is, for n ≥ 1, to discretize H into a finite grid z n, , = 1, . . . , M n , and hope to control the oscillations between the corresponding (2f (z n, )nh

d n log(1/h d n )) -1/2 T n (•, h n , z n, ). For some δ > 0, we choose z n, , = 1, . . . , M n such that ∀z ∈ H, ∃ ≤ M n , || z -z n, || d ≤ δh n , (3.12)
hoping to control the oscillations between the grid. Note that we can choose

M n ∼ λ(H)(δh n ) -d in (3.12).
By making use of, e.g., the ULDP of Proposition 2.1.2, with v k (h n ) := log(1/h d n ) we obtain, for some η > 0

P Mn =1 T n (•, h n , z n, ) 2f (z n, )nh d n log(1/h d n ) / ∈ S G,1 ≤M n max =1,...,Mn P T n (•, h n , z) 2f (z n, )nh d n, log(1/h d n, ) / ∈ S G,1 ≤λ(H)(δh n ) -d exp -(1 + η) log(1/h d n ) =O(h dη n ).
Since log(1/h n )/ log log(n) → ∞, that sequence is summable along any subsequence of the type n k ∼ (1 + γ) k (the blocking arguments then play their role). We now turn to controlling the oscillations between the z n, . Such a global oscillation is described by the following probabilities, for which it would be sufficient to prove that (if δ > 0 is sufficiently small) those are O(h η n ), a n → ∞, for some η > 0.

P max =1,...,Mn sup ||z-z n, || d ≤δhn T n (•, h n , z) -T n (•, h n , z n, ) G 2f (z n, )h d n log(1/h d n ) ≥ (3.13) ≤P sup g∈G, z∈H, ||z -z|| d ≤δhn n i=1 g hn,z (Z i ) -E g hn,z (Z i ) ≥ 2 inf z∈H f (z)nh d n log(1/h d n ) (3.14) ≤P G n (•) F hn,δ ≥ 2 inf z∈H f (z)h d n log(1/h d n ) F hn,δ := g hn,z (•) -g hn,z (• + vh n ), g ∈ G, z ∈ H, || v || d ≤ δ .
Since F satisfies (Contin. dilatations), and by a usual change of variable, the class F hn,δ satisfies

sup g∈F hn,δ Var g(Z 1 ) ≤ r(δ)h d n ,
where lim δ→0 r(δ) = 0 and where r does not depend upon n. In view of the concentration inequality of Talagrand (Fact 2.1.2, with t ∼ nh d n log(1/h d n )), appropriate exponential bounds will be achieved if it is proved that

E G n (•) F hn,δ ≤ r (δ) h d n log(1/h d n ). (3.15)
where lim δ→0 r (δ) = 0. Mason used assumption (Euclidian) to prove (3.15). Very roughly speaking, since sup

Q probab N ( , F, || • || Q,2 ) ≤ C 0 -v 0
for all > 0, we have, for some universal constants C and C :

h d 0 log sup Q probab N ( , F, || • || Q,2 )d ≤ C h d 0 log(1/ ) ≤ C h d log(1/h d ),
for small values of h. All the subtilties to obtain a proper bound for (3. To sum up : inequality (3.14) involves suprema of empirical processes over a class of functions where the point z is not fixed, but varies on the whole set H, (contrarily to local functional limit laws). Hence it is impossible to handle the whole class through a local representation such as (2.17), and hence we have to consider empirical processes over classes of function of the form g h,z , instead if the class of G directly. Moreover, without a local representation at hand such as (2.17), the considered empirical processes are built on sample of size n instead of na(h, z) ∼ f (z)nh d , which introduce another difficulty for controlling their first moments. That difficulty was overcome by Mason by strengthening (U nif. entropy dilatations) to (Euclidian). Now, if we go back to (3.13), we can also bound

P max =1,...,Mn sup ||z-z n, || d ≤δhn T n (•, h n , z) -T n (•, h n , z n, ) G 2f (z n, )h d n log(1/h d n ) ≥ 3.2. UNIFORM-IN-BANDWIDTH FUNCTIONAL LIMIT LAWS ≤M n max =1,...,Mn P sup ||z-z n, || d ≤δhn T n (•, h n , z n, ) -T n (•, h n , z n, ) G 2f (z n, )h d n log(1/h d n ) ≥ =M n max =1,...,Mn P T n (•, h n , z) G +,δ 2f (z n, )h d n log(1/h d n ) > (3.16)
By this simple remark, we see that each of the M n probabilities can be handled by a local representation around each z n, . More precisely, it is possible to use Proposition 2.1.3 uniformly in z n, , taking u := log(1/h d n ). Since Proposition 2.1.3 requires (U nif. entropy), it is clear that, under (U nif. entropy translations), we have :

M n max =1,...,Mn P T n (•, h n , z) G +,δ 2f (z n, )h d n log(1/h d n ) > ≤λ(H)δ -d h d(-1+ ∆ -2 G +,δ
) n (3.17) By (Contin. translations) that last bound can be O(h η n ), as n → ∞, for some η > 0 if δ is chosen small enough in the construction of the z n, . This concludes my explanation on how to relax (Euclidian).

Inner part

I will now give the idea on how to prove the inner part of Fact 3.2.1. It is sufficient to prove that, for arbitrary ψ ∈ S G,1 the probabilities

P z∈H T n (•, h n , z) 2f (z)nh d n log(1/h d n ) / ∈ ψ
are summable in n (no subsequence argument is involved here). The idea to prove that summability is that Θ global n (h n , H) contains an increasing number of "almost mutually independent" processes. Contrarily to what happens in Theorem 7, the independence does not happens between bandwidths, but between the points z. Suppose, that instead of local empirical processes, we were studying their Poissonized versions (this is possible by the usual Poissonization of Einmahl, see, Proposition 2.2.3). Then, by (Support), if two points z, z have distance || z -z || d ≥ 2M h n , then, by Proposition 2.2.1, the processes Π n (•, h n , z) and Π n (•, h n , z ) are rigorously independent. This can be generalized to finite collections of points z n, which are at distance at least || z -z || d ≥ 2M h n one from each other. Since H has nonempty interior, it is possible construct a number M n ∼ λ(H)h -d n of such points. By independence we then have, writing J G (ψ ) = 1 -2η, with η > 0 :

P z∈H Π n (•, h n , z) 2f (z)nh d n log(1/h d n ) / ∈ ψ ≥P Mn =1 Π n (•, h n , z n, ) 2f (z n, )nh d n log(1/h d n ) / ∈ ψ = Mn =1 1 -P Π n (•, h n , z n, ) 2f (z n, )nh d n log(1/h d n ) ∈ ψ ≤ exp -M n min P Π n (•, h n , z n, ) 2f (z n, )nh d n log(1/h d n ) ∈ ψ ≤ exp -λ(H)h -d n h d(1-η) n by a ULDP ≤ exp -λ(H)h -dη n , which is summable, since log(1/h n )/ log log(n) → ∞.

The asymptotic independence phenomenon in this framework

The first work of my PhD thesis was to generalize Fact 3.2.1 to couples (T n (•, h n , z), T n (•, h n , z)), with h n = o(h n ), generalizing a result of Deheuvels [START_REF] Deheuvels | Strong approximation of quantile process by iterated Kiefer processes[END_REF] for the uniform empirical process. This result was not sent for publication. Before stating it, I will need the definition of another type of Strassen set, given α ∈]0, 1[ :

S ⊗2 G (α) := (ψ 1 , ψ 2 ) ∈ ∞ (G) × ∞ (G), J G (ψ 2 ) ≤ 1, J G (ψ 1 ) + αJ G (ψ 2 ) ≤ 1 .
Also define, for n ≥ 1 and for two bandwidths h n ≤ h n :

Θ global,⊗2 n (h n , h n , H) := T n (•, h n , z) 2f (z)nh d n log(1/h d n ) , T n (•, h n , z) 2f (z)nh d n log(1/h d n ) , z ∈ H .
Theorem 8 (Varron, 2003, [94]) Assume that the law of Z 1 fulfills (Hf ) for a set O. Let H ⊂ O be a compact set with nonempty interior. Assume that G fulfills (P ointw. sep.), (Bounded), (Support), (U nif. entropy dilatations), (U nif. entropy translations), (Contin. translations) and (Contin. dilatations). Let h n ≤ h n both fulfilling the Csörgő-Révész-Stute conditions :

h n ↓ 0, nh d n ↑, nh d n log(1/h d n ) → ∞, log(1/h n ) log log(n) → ∞.
there are still h

-d(1-α+J(ψ 1 )) n independent possibilities that (2f (z n, )nh d n log(1/h d n )) -1/2 T n (•, h n , z n,
) hits a target ψ 1 , which allows ψ 1 to have a non negligible stochastic cost, less than 1 -α.

In bandwidth uniformity for the global standard limit law

My second main result concerning in-bandwidth uniformity is related to global standard functional limit laws for the local empirical process.

Theorem 9 (Varron, 2008 [98]) Let h n ≤ h n be two bandwidths sequences fulfilling (3.12). Then, under the assumptions of Fact 3.2.1, we have, almost surely

lim n→∞ sup h∈[hn,hn] d Haus Θ global n (h, H), S G,1 = 0.
Later Deheuvels and Ouadah [START_REF] Deheuvels | Uniform-in-Bandwidth Functional Limit Laws[END_REF] did remove condition log(1/h n )/ log log(n) → ∞, for the uniform empirical process, at the price of relaxing the almost sure convergence to a convergence in probability. Sketch of the proof and comments : Let us consider the outer part of the proof. As I tried to emphasize in a brief description of the proof of Fact 3.2.1, one of the key aspects of the problem is that, given a sequence h n fulfilling (3.12), the probability of

E n ( , h n ) := Θ global n (h n , H) -S G,1 = ∅ is (for fixed > 0) of order h η
n , for some η > 0. Now, proving (3.18) would necessarily require a discretization of the intervals [h n , h n ]. As it was already shown in the proof of Theorem 7, assumptions (U nif. entropy dilatations) and (Contin. dilatations) allow to control the oscillations between bandwidths of the form [h, ρh], as soon as ρ > 1 is small enough. We will therefore define, for fixed ρ > 1, the grid h n, := ρ h n , ≤ R n , with R n ≤ log(h n /h n )/ log(ρ). Now, using the ULDP of Proposition 2.1.2, with ñk := n and v k (h) := log(1/h d ) it is possible to uniformly control the probabilities of E n ( , h n, ), ≤ R n , and hence use Bonferroni's inequality

P Rn =0 Θ global n (h n, , H) -S G,1 = ∅ ≤ Rn =0 P E n ( , h n, ) ≤ Rn =0 h η n, =h η n Rn =0 ρ η 68 3.2. UNIFORM-IN-BANDWIDTH FUNCTIONAL LIMIT LAWS ≤h η n ρ η(Rn+1) ρ η -1 = ρ η ρ η -1 h η n exp δR n log(ρ) = ρ η ρ η -1 h η n .
Which is of the same order as if we had taken h n = h n . The same arguments hold for the inner part of the proof. Hence, contrarily to what happens in the local standard functional limit law, summing all these probabilities does not hurt at all, since those probabilities form a geometric sequence. In other words the probability of E n ( , h n ) is preponderant in front of the others, which roughly speaking means that sup h∈[hn,hn]

d Haus Θ global n (h), S G,1 d Haus Θ global n (h n ), S G,1 .

Perspectives

My main perspective is to investigate how these functional limit laws can be extended to classes that fail to be uniformly bounded. This perspective is completely embedded in the perspective of improvements of the ULDP of the preceding chapter (see §2.1.6).

A nonfunctional uniform-in-bandwidth limit law for an object related to nonparametric regression

Several techniques developed for the local empirical processes can be used out of the context of functional limit laws. This is typically the case for concentration inequalities coupled with control of first moments, as it was pointed out by Einmahl and Mason [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF]. In this article, they could prove the exact almost sure rates of convergence for objects related to nonparametric density/ regression estimation, which is described as follows. Given an i.i.d sample (Y i , Z i ) i=1,...,n taking values in R d × R d , with the same distribution as a vector (Y, Z), we assume that the law of (Y, Z) satisfies (Hf ) for some product of open sets

O = O 1 × O 2 , with O 1 ⊂ R d and O 2 ⊂ R d . Now write V n (g, h, z) :=f Z (z) -1/2 n i=1 c g (z) • g(Y i ) + d g (z) K h,z (Z i ) -E c g (z) • g(Y i ) + d g (z) K h,z (Z i ) , (3.18)
where

-K denotes a kernel -h > 0 is a bandwidth -g is a Borel function from R d to R k -f Z is (a version of) the density of Z -c g maps R d to R k and d g maps R d to R -The symbol • stands for the canonical inner product on R k .
As pointed out by Einmahl and Mason [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF], those objects are of statistical interest, since they encompass the main stochastic term of several estimators, ranging from the Parzen-Rosenblatt density estimator to the kernel estimator of the conditional distribution functions. Given a class G, of functions from R d to R k , we will write, for = 1, . . . , k the class G := π (G), where π (x 1 , . . . , x , . . . , x k ) := x for (x 1 , . . . , x k ) ∈ R k . The following structural assumptions are commonly fulfilled in practice.

(V Csubgraph) Each class G is a pointwise separable VC subgraph, (Envelope)
Each class G has a finite valued measurable envelope function G satisfying, for some p ∈ (2, ∞] :

max =1,...,k sup y∈O1 || G (•) || L Y |Z=z ,p < ∞, where || G (•) || L Y |Z=z ,p is the L p -norm of G under the distribution of Y | Z = z (which is uniquely defined if we use the continuous version of f Y,Z ).
For a definition of a VC subgraph class, we refer, e.g., to [92, p.141].

The assumptions made upon the collections c g (•) g∈G and d g (•) g∈G are as follows.

(HC) The classes of functions D 1 := {c g , g ∈ G} and D 2 := {d g , g ∈ G} are uniformly bounded and uniformly equicontinuous on O 2 .

The assumption on the kernel K, involves the class.

K := K h,z , h > 0, z ∈ R d .
(HK1) K has bounded variation and the class K is VC subgraph, (HK2) (3.18)), along a bandwidth sequence (h n ) n≥1 that satisfies the following conditions

K(s) = 0 when s / ∈ [-1/2, 1/2] d , (HK3) R d K(s)ds = 1.

Einmahl and Mason have studied the almost sure asymptotic behavior of sup

z∈H ∆ n (G, h n , z), with ∆ n (G, h, z) := sup g∈G | V n (g, h, z) | 2f (z)nh d log(1/h d ) (recall
(HV ) h n ↓ 0, nh n d ↑ ∞, log(1/h n )/ log log(n) → ∞, h d n n/ log(1/h n ) 1-2/p → ∞,
where p is as in condition (Envelope).Let us define

∆ 2 (g, z) :=E < c g (z), g(Y ) > +d g (z) 2 | Z = z , z ∈ R d , g ∈ G, (3.19)
(Y, Z) admits a density. That condition is somewhat unsatisfactory, because the smoothing is only made on the Z i .

A third tough challenge is to establish Bickel-Rosenblatt limit laws for such estimators, namely proving that, for some sequence

a n , b n → ∞ a n sup h∈[hn,hn] sup z∈H ∆ n (G, h, z) -∆(G, z) || K || λ,2 -b n → L U,
where U is a probability distribution on R. Obtaining such results would have a more significant practical impact that the functional limit laws, since a convergence in distribution may allow calibrations. Such a limit law was proved by Bickel and Rosenblatt [START_REF] Bickel | On some global measures of the deviations of density function estimates[END_REF] for the Parzen-Rosenblatt density estimator, for univariate distributions (i.e. G = 1 , d = 1 and

h n = h n ).
In that case

a n = log(1/h d n ) and b n = log log(1/h n ).
In their proof they make use of two crucial tools -The KMT strong approximation.

-The analysis of the distributions of suprema of Gaussian processes over increasing intervals, following ideas of Pickands [START_REF] Pickands | Upcrossing probabilities for gaussian processes[END_REF].

It is essentially the KMT approximation that prevents a generalization to d ≥ 3 of their methods. I am currently working on using the modern tools in empirical processes theory (concentration inequalities and approximations results of Zaitsev) to extend such a limit law to d ≥ 3. I however, came across to a stumbling block, which can be described as follows.

-Poissonization works, so we can directly study the Π n (1, h n , z).

-Such a level of precision imposes the analysis of max =0,...,Mn

Π n (1, h n , z n, ), (3.21)
where, in the definition of the z n, , we choose δ n ∼ δ log(1/h d n ) -1 instead of δ > 0 (as in 3.12).

-The joint law of those M n random variables cannot be approximated by the joint law of Gaussian random variables : the dimensionality explodes with n, which makes Zaitsev's bounds unusable. However, it is possible to make a sharp enough approximation separately on subblocks of cardinalities log

(1/h d n ). This defines δ -d h -d n log(1/h d n ) sub blocks.
The distances between two consecutive sub blocks is δh n , with δ small. Hence, there is a non negligible dependency between the blocks. -In their proof, Bickel and Rosenblatt could directly work on a joint Gaussian law for all those blocks (thanks to the KMT approximation). Even if those Gaussian blocks are not mutually independent, the authors could work on the covariance structure of these random variables, and conclude to a convergence of these maxima by making use of results of Pickands [START_REF] Pickands | Upcrossing probabilities for gaussian processes[END_REF]. -Without a strong approximation, I do no know how to translate their Gaussian analysis (a control of the covariance structure) to a Poisson analysis. The underlying reason is that, for f gdλ = 0 implies W G (f ) ⊥ ⊥ W G (g), while, for a Poisson process, such an independence is guaranteed only when f and g have disjoint supports.

Spatial-type limit laws

This section is dedicated to another type of functional limit laws (as well as a Donsker theorem) which are different from the usual functional limit laws for local empirical processes, in the sense that they are spatial.

A spatial Donsker theorem

First results for the uniform empirical process Wschebor [START_REF] Wschebor | Sur les accroissements du processus de Wiener[END_REF] discovered the following property of the Wiener process W on [0, 1] : almost surely, for each 0 ≤ a < b < 1, and for each Borel set B ⊂ R,

1 (b -a) λ z ∈ [a, b], -1/2 W (z + ) -W (z) ∈ B → →0 P N (0, 1) ∈ B .
Recall that λ denotes the Lebesgue measure. That result was later extended to a much wider class of processes by Azaïs and Wschebor [START_REF] Azais | Almost sure oscillation of certain random processes[END_REF]. It can be interpreted as an almost sure spatial convergence in distribution. Here, the word "spatial" refers to the fact that it involves the almost sure limits of (Lebsesgue) measures of random sets of points of [0, 1].

A first question of interest is to establish the same limit result for the functional increments of the Wiener process, namely

1/2 W (z + •) -W (z) ∈ ∞ ([0, 1]), (3.22)
hence replacing a limit normal distribution by a Wiener measure.

A second natural question is to determine how this property of the Wiener process is shared by the empirical process α n (see §3.1.3). My first result in that direction [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF] is stated as follows, writing

∆ n (•, h n , z) := h -1/2 n α n (z + h n •) -α n (z) ,
as processes on [0, 1], and writing λ * for the outer Lebesgue measure on [0, 1].

Theorem 12 (From Varron, 2011 [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF]) Assume that : Strong approximation under || • || [0,1] (such as the KMT approximation) is a too heavy mathematical tool to handle this problem, since we only need to control Lebesgue measures. Hence, it is not surprising that the preceding theorem has a direct extension to the multivariate uniform empirical process on [0, 1] d , as it is stated in its complete form in [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF].

h n ↓ 0, nh n ↑ ∞, lim n→∞ log(1/h n )/ log log(n) > 1. ( 3

Sketch of the proof and comments

In his original result, Wschebor used the property of independence of increments of the Wiener process. His proof is decomposed as follows : By a separability argument, we only need to prove that, for any fixed bounded Lipschitz function φ, from R to R, we have

lim →0 Z(φ, ) = a.s. E φ(Z) with Z(φ, ) := 1 λ(I) I φ √ W (z + ) -W (z) dz,
and with Z denoting a standard normal random variable.

-(Expectation) The expectation of the random variables of interest is simply computed as

E Z(φ, ) = 1 λ(I) I E φ √ W (z + ) -W (z) dz =E φ Z .
Hence, the almost sure limit to establish is simply the expectation of the Z(φ, ), which does not depend upon . -(V ariance) Now the independence of increments plays its role in the computation of the related variance (equality (3.24) below) :

Var Z(φ, ) = 1 λ(I) 2 I 2 Cov φ √ W (z + ) -W (z) , φ √ W (z + ) -W (z ) dzdz = 1 λ(I) 2 |z-z |≤ Cov φ √ W (z + ) -W (z) , φ √ W (z + ) -W (z ) dzdz (3.24)
=O( ), as → 0.

-(Interpolation) The preceding variance bound proves the required almost sure convergence along each sequence n = n -(1+δ) , with δ > 0. Then, the almost sure Hölder property of W ensures the interpolation between the n .

Those three key arguments can be extended with no efforts to obtain the same limit result for the functional increments in (3.22).

The idea of the proof of Theorem 12 is to use similar arguments for the ∆ n (•, h n , z).

-

Z n (φ, h n ) := 1 λ(I) I φ ∆ n (•, h n , z) dz, (3.25)
we have

E Z n (φ, h n ) = 1 λ(I) I E φ ∆ n (•, h n , 0) dz =E φ ∆ n (•, h n , 0) .
But, since nh n → ∞, Donsker's theorem for the local uniform empirical process (see [START_REF] Mason | A strong invariance principle for the tail empirical process[END_REF]) states that

lim n→∞ E φ ∆ n (•, h n , 0) = E φ W (•) ,
which provides an adapted counterpart of (Expectation). Note that the preceding arguments are not rigorous, since φ ∆ n (•, h n , 0) may fail to be Borel measurable. The proof can nevertheless be handled by considering only a subclass of measurable functions φ, involving evaluations at a countable number of elements of G. However, for concision, I will skip all the technicalities used to tackle such a lack of measurability, and state all the subsequent arguments "as if φ ∆ n (•, h n , 0) was measurable".

-The counterpart of (V ariance) is based on Poissonization, in order to take advantage of the independent increments of a Poisson process on the real line. By a simple union argument, it can be supposed that P U 1 ∈ I < 1/2 without losing generality. In that case, the random variables Z n (φ, h n ) are entirely determined by the values U i falling in I hn . In view of the Poissonization technique in Proposition 2.2.4, it is hence not surprising that one can Poissonize the variance of Z n (φ, h n ) (beside some small technicalities that are voluntarily skipped -for more details, see [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF]Proposition 3.2]). This leads to the following calculus, writing

∆Π n (•, h, z) := ηn i=1 1 [z,z+h•] (U i ) -h • √ nh ,
where η n is a Poisson random variable with expectation n and independent of (U i ) i≥1 .

Var I φ ∆Π n (•, h n , z) dz = I 2 Cov φ ∆Π n (•, h n , z) , φ ∆Π n (•, h n , z ) dzdz =O(h n ),

Extension to more general local empirical processes

The preceding results was written here for the concision and clarity of the main arguments of its proof. I could, however, generalize it to the wider class of objects T n (•, h n , z), under assumptions that are very similar to those used in the results of §3.2. I could also use the same trick as in the proof of Theorem 9, to enrich the generalization with an additional uniformity in h ∈ [h n , h n ].

Theorem 13 (From Varron, 2014, [START_REF] Varron | The almost sure behavior of certain spatial distributions of local empirical processes indexed by functions[END_REF]) Assume that Z 1 satisfies (Hf ) for some O. Let H be a set with nonempty interior fulfilling H ⊂ O (H denoting the closure of H). Let µ be a continuous probability measure on H having a version of Lebesgue density which is uniformly bounded. Let h n and h n be two bandwidths sequences fulfilling h n ↓ 0, nh d n ↑ ∞, as well as

lim n→∞ log(1/h d n )/ log log(n) > 1.
Assume that G fulfills (U nif. entropy dilatations), (Contin dilatations), (P ointw. sep.) (Support), and admits an envelope that is square integrable under the Lebesgue measure. Then the following assertion holds with probability one : for each closed set

F of ∞ (G), || • || G , we have lim n→∞ sup h∈[hn,hn] µ * z ∈ H, T n (•, h, z) f (z)nh d ∈ F ≤ P W G ∈ F ,
where W G is the isonormal Gaussian process spanned by G, || • || λ,2 .

Note that the preceding theorem holds under conditions that are slightly weaker that those written here, but somewhat cumbersome to expose. For more details see [START_REF] Varron | The almost sure behavior of certain spatial distributions of local empirical processes indexed by functions[END_REF].

Sketch of the proof an comments

Write, for a bounded Lipschitz function φ :

Z n (φ, h) := H φ T n (•, h n , z) f (z)nh d n dµ(z),
ignoring measurability issues for simplicity of the presentation.

-The usual blocking arguments still hold for the general objects

T n (•, h, z), h ∈ [h n , h n ], z ∈ H.
-

(z, z ) ∈ H 2 , || z -z || d ≤ 2M h is O(h d ) as h → 0. Since h n → 0 we obtain Var Z n (φ, h) ≤ Ch d , uniformly in h ∈ [h n , h n ],
for a constant C, and for n large enough. In addition, since the involved variances are of order h d , it is possible to use the discretization procedure h n, = ρ h n , ≤ R n and sum up the corresponding variances to obtain a sum

Rn =0 Var Z n (φ, h n, ) ≤ C ρ d -1 h d n . (3.26)
Hence requiring a uniformity in h n, does not hurt. The interpolation arguments between consecutive h n, are standard, and involve (U nif. entropy dilatations) and (Contin. dilatations).

-It remains to prove an adapted counterpart of the (Expectation) arguments. Consider the problem of proving

E φ W G = lim n→∞ E I φ T n (•, h n , z) f (z)nh d n dµ(z) = lim n→∞ I E φ T n (•, h n , z) f (z)nh d n dµ(z). (3.27)
The law of T n (•, h n , z) is not constant in z ∈ H, as it was the case for the uniform empirical process. However, by the dominated convergence theorem (under the measure µ) it is sufficient to prove that

lim n→∞ E φ T n (•, h n , z) f (z)nh d n = E φ W G ,
for µ-almost each z. That convergence is simply Donsker's theorem for local empirical processes, which is a consequence of Einmahl and Mason [START_REF] Einmahl | Gaussian approximation of local empirical processes indexed by functions[END_REF].

Because our result involves a uniformity in h ∈ [h n , h n ], such a uniformity has to be added in (3.27), and hence raises additional technicalities that will not be detailed in this manuscript. For more details, see [START_REF] Varron | The almost sure behavior of certain spatial distributions of local empirical processes indexed by functions[END_REF].

Spatial functional limit laws

Since the preceding results are spatial-type Donsker theorems, it seemed natural to me to investigate spatial-type standard/nonstandard functional limit laws. After first results concerning the uniform empirical process [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF], I could obtain the following spatial standard functional limit law, which is the spatial counterpart of Theorem 7.

Theorem 14 (Varron, 2014 [104]) Assume that the law of Z 1 satisfies (Hf ) for some open set O. Let H ⊂ O be a compact set and let µ be a probability measure on H. Let h n ≤ h n be two non random sequences fulfilling :

(HV ) nh d n ↑, h n ↓ 0, nh d n / log log(n) → ∞, lim n→∞ log log(h n /h n ) log log(n) = δ ∈]0, 1].
Then, under assumption (Bounded), (Support), (P ointw. sep.), (U nif. entropy dilatations), (Contin. dilatiations), (U nif. entropy translations), (Contin. translations), the following assertions hold with probability one :

P µ z ∈ H, top lim n→∞ Θ local n (h n , h n , z) = S 1+δ G = 1 = 1;
(3.28)

3.3. SPATIAL-TYPE LIMIT LAWS P µ z ∈ H, top lim n→∞ Θ local n (h n , h n , z) = S δ G = 1 = 1.

Sketch of the proof and comments

The arguments of §3.3.1 show that, for particular bounded functions φ, we have almost surely Z n (φ, h n ) → E φ W G . A functional limit law relies of large deviations, which provide rates of convergence to zero of probabilities of rare events. Hence, my first idea was to replace the function φ by sequences of functions φ n of indicators of rare events (i.e. φ n T n (•, h n , z) → 0 almost surely) and see how the (Expectation) and (V ariance) arguments could be translated in this context. This work took form in a first result for the uniform empirical process (see [START_REF] Varron | Some new almost sure results on the functional increments of the uniform empirical process[END_REF]). However, that methodology imposed the condition

lim n→∞ log(1/h d n ) log log(n) > 2,
which was required to obtain better rates of approximations between Z n (φ n , h n ) and their expectations, in order to have

Z n (φ n , h n ) = O a.s. E Z n (φ n , h n ) , (3.29)
the latter tending to zero as n → ∞. With approximation results such as (3.29), I could then obtain almost sure estimates for Z n (φ n , h n ) which are "almost sure µ-large deviations". I then used these large deviations under µ, coupled with a blocking argument to obtain such a spatial functional limit law. In a sense, I did my best to mimic the arguments of the usual Strassen laws, adapting them to a spatial setting. Afterwards, when I tried to generalize my result to the more general objects T n (•, h n , z), I realized that I did not actually need to mimic those arguments. Indeed, there is a different way to prove it, by noticing that some crucial random sequences are almost surely monotonic, and hence only their convergences in probability need to be proved. Let us illustrate this through one particular step of the proof. To prove the inner part of (3.28), we need to prove that

µ n 0 ≥1 n≥n 0 h∈[hn,hn] z ∈ H, T n (•, h, z) 2f (z)nh d log log(n) / ∈ ψ = 0, a.s.
Since, for each n 0 , the sequence

µ n 0 +p n=n 0 h∈[hn,hn] z ∈ H, T n (•, h, z) 2f (z)nh d log log(n) / ∈ ψ .
is almost surely decreasing in p, it is sufficient to prove the convergence to 0 (as p → ∞) of

E µ n 0 +p n=n 0 h∈[hn,hn] z ∈ H, T n (•, h, z) 2f (z)nh d log log(n) / ∈ ψ ≤ sup z∈H P n 0 +p n=n 0 Rn =0 T n (•, h n, , z) 2f (z)nh d n, log log(n) / ∈ ψ ,
where we introduced a sufficiently spread grid h n, := ρ n h n , ≤ R n , with ρ n → ∞. By the usual stochastic renewal argument, the first intersection symbol can be converted to a product of probabilities of independent events (along sufficiently spread subsequences n k ), this leads to the following heuristic :

sup z∈H P n 0 +p n=n 0 Rn =0 T n (•, h n, , z) 2f (z)nh d n, log log(n) / ∈ ψ ≤ exp -inf z∈H n 0 +p n=n 0 P Rn =0 T n (•, h n, , z) 2f (z)nh d n, log log(n) ∈ ψ . (3.30)
Now the idea is to use the same arguments as those used to prove the heuristic (3.9), with the formal replacement of z 0 by z. An additional uniformity in z ∈ H is required though, but that uniformity is perfectly handled by the ULDP and the Poissonization techniques of Propositions 2.1.2 and 2.2.2. The obtained lower bounds are large enough to ensure the required convergence to 0, along the sequence n k used in the stochastic renewal argument. We can see that, using those kind of arguments, it is not needed to have precise almost sure estimates for random sequences of the form Z n φ n , h n, . For example, looking at (3.30), it is sufficient to obtain lower large deviations bounds for T n (•, h n, , z), which hold uniformly in z ∈ H and h n, , ≤ R n . Since precise estimates are not required, so are the assumption on lim n→∞ log(1/h n )/ log log(n), and so is the assumption that µ has a bounded density, because the (V ariance) step can be bypassed. .

Perspectives

As already mentioned for problems involving the local empirical process, an important perspective would be to relax the (Bounded) assumptions to high-order integrability of the envelope. Another perspective is as follows : if we compare the assumptions in Theorems 13 and 14, we can see that the second does not require µ to be continuous, nor any condition upon log(1/h n )/ log log(n). It is somewhat surprising because functional limit laws usually require stronger assumptions than Donsker theorems. I will continue my investigations in the direction of removing those two conditions. Note, however, that Theorem 13 automatically excludes discrete measures with finitely many atoms, because le limit measure takes all possible values in [0, 1]. A crucial perspective is to investigate the possible connections between those spatial limit laws and some statistical procedures.

Chapter 4

Contributions to the study of the general empirical process, and its applications to statistics

In this chapter, the central object will be the general empirical process, namely :

G n (•) := g → √ n P n (g) -P 0 (g) ,
as defined in the introduction. My works on that object can be decomposed into three parts :

-In Bayesian nonparametrics, several priors can be seen as random probability measures that are almost surely discrete. An almost surely discrete random probability measure can be seen as Assume that (Ω, A, P) is a complete probability space (in the sense that every P-negligible set belongs to A), and let

n≥1 β i δ Y i ,
S := p = (p i ) i≥1 ∈ [0, ∞[ N , || p || c,1 < ∞ ,
be the cone of positive summable sequences, which will be endowed with the product Borel σalgebra (denoted by Bor). Consider a sequence of S-indexed collections of probability measures P n,p n≥1,p∈S on a measurable space X, X . Note that those P n,p are not random. For fixed n, consider a random variable β n = (β i,n ) i≥1 from (Ω, A, P) to (S, Bor) and a sequence Y n = Y i,n i≥1 of random variables from (Ω, A, P) to X, X , for which the conditional law given β n = p is P ⊗N n,p (we also make the assumption that those conditional laws properly define a Markov kernel). We then define the random element

P r n := i≥1 β i,n δ Y i,n ,
as a map from Ω to M (recall that the latter is the set of all probability measures on X, X ). Obtaining asymptotic results (as n → ∞) for P r n is of interest for at least two reasons. The first of them is that P r n generalizes the usual empirical measure, and also related objects from bootstrap theory, such that the empirical bootstrap, or more generally, the exchangeable bootstrap empirical measure (see [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF]Section 3.6.2]). Our main motivation to consider such a generalization comes from the second reason : these kind of random measures play a role in the representation of posterior distributions in Bayesian nonparametrics. The simplest can be described as follows : endow M with the Borel σ-algebra M spanned by the weak topology. Given a probability measure α on X, X and a concentration parameter M > 0, the Dirichlet process distribution DP (M, α) on M, M admits the following representation, due to Sethuraman [START_REF] Sethuraman | A constructive definition of dirichlet measures[END_REF] :

DP (M, α) = d i≥1 β i δ Y i .
Here, the Y i are i.i.d with common distribution α and

β i := U i i-1 j=1
(1 -U j ), with (U i ) i≥1 being an i.i.d. sample, independent of (Y i ) i≥1 , and having the Beta(1, M ) distribution. It is also well known [44, Section 3, Theorem 1] that, if a random probability measure P r has distribution DP (M, α) and if (X 1 , . . . , X n ) has law P ⊗n given P r = P (for P r almost all P), then a version of the posterior distribution of P r given (X 1 , . . . , X n ) is the map

(x 1 , . . . , x n ) → DP M + n, θ n α + (1 -θ n )α (x 1 ,...,xn) ,
where θ n := M/(M + n) and α (x 1 ,...,xn) := n -1 i≤n δ x i . A more general class of random measures admitting such a representation is the class of normalized homogenous completely random measures with no fixed atoms (see, e.g., [49, p. 84-85]).

A Glivenko-Cantelli theorem

We will consider a class of functions F which satisfies (P ointw. sep.). We will write F for its minimal measurable envelope. For r > 0, we define the space E F ,r as follows : a map Ψ : Ω → R F belongs to E F ,r if and only if Ψ(f ) defines a Borel random variable on (Ω, A, P) for each f ∈ F, and if

||| Ψ ||| r F ,r := E || Ψ || r F < ∞. Under the assumption that E F (Y 1,n ) < ∞ for all n, it is possible to define the process G Yn,β n (•) : f → i≥1 β i,n f (Y i,n ) -E f (Y i,n ) | β n as the limit, as k → ∞, of the truncated sequence G k Yn,β n (•) : f → k i=1 β i,n f (Y i,n ) -E f (Y i,n ) | β n in the Banach space E F ,1 , ||| • ||| F . Note that this limit also holds in E F ,r for each r ≥ 1 such that E F (Y 1,n ) r < ∞. My first contribution is a Glivenko-Cantelli theorem.
Theorem 15 (Varron, 2014, [105]) Assume that || β n || c,1 = 1 almost surely for all n, and

|| β n || c,2 → 0 in probability. Suppose that lim M →∞ lim n→∞ E F (Y 1,n )1 {F (Y 1,n )≥M } = 0. (4.1)
Also assume that, for each > 0 and M > 0, we have, as n → ∞ :

log N , F M , || • || P (β n ,Yn),1 = o P || β n || -2 c,2 ,
where, F M := f 1 {F ≤M } , f ∈ F and

P (p, y) := i≥1 p i δ y i , for p ∈ S, y ∈ X N . Then E || G Yn,β n || F → 0. (4.2)
Proof : The proof of 15 is an adaptation of the usual symmetrization and conditioning arguments for Glivenko-Cantelli theorems (see, e.g., [92, p. 123]). No consequent technical difficulty does show up along these steps. Note that, choosing β i,n := n -1 when i ≤ n and 0 otherwise leads to the usual Glivenko-Cantelli theorem under random entropy conditions (see, e.g., [92, p.123]), except for the almost sure counterpart of (4.2). Indeed, that almost sure convergence deeply relies on a reverse submartingale structure, which is not guaranteed under the general conditions of Theorem 15.

A Donsker theorem

My second result is a Donsker Theorem. Since, for fixed n, the Y i,n are only conditionally independent, such a result will not involve the Gaussian analogues of the G Yn,β n , but mixtures W Yn,β n of the P n,p -Brownian bridges by β n . Such mixtures of Gaussian processes can fail to be measurable with respect to the Borel σ-algebra of ∞ (F), || • || F . This lack of measurability will be tackled by introducing outer expectations (see, e.g. [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF]Chapter 1.2]). In the next theorem, the E * symbol, when applied to G Yn,β n , denotes the outer expectation taking (Ω, A, P) as underlying space. When applied to W Yn,β n , that symbol should be understood as taking the canonical probability space R F , endowed with the product Borel σ-algebra, and endowed with the probability measure exhibited by Kolmogorov's extension theorem. For more details, see [START_REF] Varron | Donsker and glivenko-cantelli theorems for a class of processes generalizing the empirical process[END_REF].

Theorem 16 (Varron, 2014, [105]) Assume that, for each n ≥ 1, || β n || 2 c,2 = 1 with probability one, and that || β n || c,4 → 0 in probability. Also assume that E F 2 (Y 1,n ) < ∞ for all n, and that :

lim M →∞ lim n→∞ E F 2 (Y 1,n )1 {F (Y 1,n )≥M } = 0, ∞ 0 log sup Q probab. N || F || Q,2 , F, || • || Q,2 d < ∞.
Then, for all n ≥ 1, W Yn,β n is almost surely bounded, and || W Yn,β n || F is Borel measurable.

A DONSKER AND A GLIVENKO-CANTELLI THEOREM FOR A CLASS OF RANDOM MEASURES RELATED TO THE EMPIRICAL MEASURE

Also assume that, for a semimetric ρ 0 that make F totally bounded we have :

lim δ→0 lim n→∞ P * sup (f 1 ,f 2 )∈F 2 ,ρ 0 (f 1 ,f 2 )≤δ || f 1 -f 2 || P n,β n ,2 = 0, (4.3)
where the symbol lim P * n→∞ stands for the lim sup in outer probability. Then

d BL G Yn,β n , W Yn,β n := sup φ∈BL1 E * φ G Yn,β n -E * φ W Yn,β n → 0,
where BL1 is the set of all 1-Lipschitz functions on ∞ (F), || • || F that are bounded by 1.

Remark : When (Y i,n ) i≥1 ⊥ ⊥ β n , condition (4.
3) is implied by condition (ii) in Sheehy and Wellner [83, Theorem 3.1], namely, for some probability measure P 0 :

sup (f,g)∈F 2 max | P (n) 0 (f -g) 2 -P 0 (f -g) 2 |, | P (n) 0 -P 0 (f ) | | P (n) 0 (f 2 ) -P 0 (f 2 ) | → 0, (4.4)
where P

(n) 0 stands for the law of Y 1,n . In this particular case, Theorem 16 turns out to be a partial generalization of Sheehy and Wellner [START_REF] Sheehy | Uniform donsker classes of functions[END_REF]Theorem 3.1], where the authors proved, among other results, a Donsker theorem for sequences of F-indexed empirical processes, for which the law of the sample varies with n. The main advantage of our result is that it extends to random (possibly infinite) convex combinations of the δ Y i,n . As a consequence of Theorem 16, I could obtain an alternate proof of the Bernstein-Von Mises theorem under || • || F -topologies, for the Dirichlet process prior. Such a result was already obtained by James [START_REF] James | Large sample asymptotics for the two-parameter poisson-dirichlet process[END_REF], under the slightly more stringent assumption that F is VC subgraph.

Sketch of the proof and comments

The measurability of || W Yn,β n (•) || F uses the fact that F is pointwise separable in combination with the dominated convergence theorem, which implies (under assumptions of integrability of the envelope) that there exists a countable F 0 ⊂ F which, for almost each ω ∈ Ω , is dense in F, || • || P n,β n (ω) ,2 . The gaussian structure conditionally to β n does the rest. The proof of Theorem 16 is then decomposed into two parts :

Step 1: convergence of the marginals Write d LP for the Levy-Prokhorov distance between (Borel) probability measure on R p , generated by || • || p , namely, for two probability measures P and Q : d LP (P, Q) := inf λ > 0, π(P, Q, λ) ≤ λ , where we remind that π(P, Q, λ) := sup

A Borel max P (A) -Q(A λ ), Q(A) -P (A λ ) , for λ > 0,
The first step consists in controlling the distance of the marginals of G Yn,β n and those of W Yn,β n . Recall that Π f (ψ) = ψ(f 1 ), . . . , ψ(f p ) , as defined in (2.2). Proposition 4.1.1 For each p ≥ 1 and f ∈ R p , we have

d LP Π f (G Yn,β n ), Π f (W Yn,β n ) → 0.
The main argument of the proof of Proposition 4.1.1 is a generalization, to infinite convolutions, of the approximation results of Zaitsev [START_REF] Yu | On the Gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality[END_REF] (a weaker form of Zaitsev's result can be found in the present manuscript, in Fact 2.1.1). Such a generalization is simple since Zaitsev's bounds do not depend on the sample size.

Step 2 : asymptotic equicontinuity The second step consists in showing that (W Yn,β n ) n≥1 and (G Yn,β n ) n≥1 are asymptotically equicontinuous. First, from assumption (4.3), the the equicontinuity arguments boil down to showing that

∀ > 0, lim δ→0 lim n→∞ P sup (f 1 ,f 2 )∈F 2 , ||f 1 -f 2 || P n,β n ,2 <δ G Yn,β n (f 1 ) -G Yn,β n (f 2 ) = 0, (4.5)
∀ > 0, lim

δ→0 lim n→∞ P sup (f 1 ,f 2 )∈F 2 , ||f 1 -f 2 || P n,β n ,2 <δ W Yn,β n (f 1 ) -W Yn,β n (f 2 ) = 0. (4.6)
The fact that those two suprema are measurable can be showed by density of F 0 simultaneously for the pointwise convergence and for the norms

Perspectives

A commonly used class of nonparametric discrete priors is the class of normalized completely random measures. A normalized homogenous completely random measure admits the representation (4.1), with the Y i,n being conditionally i.i.d given the weights. In [START_REF] James | Posterior analysis for normalized random measures with independent increments[END_REF], James, Lijoi and Prünster did prove a very general result, by giving the general formula for (a version of) the posterior distribution (given a sample (X 1 , . . . , X n )) of a normalized completely random measure. Such a formula does not directly imply objects such as (4.1) but mixtures of such random measures with respect to latent variables U (X 1 , . . . , X n ). A very appealing and apparently tough challenge would be to investigate the posterior consistency and rate of convergence, under || • || F of normalized homogenous completely random measures. James [START_REF] James | Large sample asymptotics for the two-parameter poisson-dirichlet process[END_REF] already showed that it is not always the case by proving general consistency/unconsistency results when considering the Poisson-Dirichlet process as a prior. When the underlying law of the sample is continuous, the Dirichlet process prior is the only member of that family that benefits of || • || F -consistency (except for trivial classes).

Empirical processes theory for empirical likelihood

The empirical likelihood principle

Empirical likelihood was first introduced by Thomas and Grunkemeier [START_REF] Thomas | Confidence interval estimation of survival probabilities for censored data[END_REF] in a setup of survival analysis. Owen ([73,[START_REF] Owen | Empirical likelihood ratio confidence regions[END_REF][START_REF] Owen | Empirical likelihood for linear models[END_REF]) then generalized their concept and initiated a fruitful theory in estimation (see [START_REF] Owen | Empirical Likelihood[END_REF] for an overview). The underlying idea is simple to explain : consider an i.i.d. sample (Y 1 , . . . , Y n ) taking values in R p , and admitting a nondegenerate variance, and consider the (most) simple problem of building confidence regions for their expectation θ 0 . Instead of building ellipsoids centered at the point estimator where u ∈ (0, 1) is a specified threshold. Owen showed that, under those simple assumptions, there is a way to calibrate u α given a specified asymptotic confidence level 1-α. More precisely, he showed the following convergence in distribution.

Fact 4.2.1 [START_REF] Owen | Empirical likelihood ratio confidence intervals for a single functional[END_REF] Assume that Y 1 admits a variance that is nondegenerate. Then Note that θ 0 ∈ R n,u ⇔ EL n ≥ u.

One of the advantages of this empirical likelihood method is that one doesn't need to estimate the variance in order to build the confidence region. Moreover, the shape of those regions strongly depends on the geometry of the data whereas the classical central limit theorem gives ellipse-shaped confidence regions. Finally, the confidence band always lies inside of the convex hull defined by the data. Since Owen's grounding result, several researchers did contribute to the extension of this method toward the estimation of parameters that are not expectations of finite dimensional data.

Estimations of linear, trajectory-valued functionals

With J.Y Dauxois and A. Flesch [START_REF] Dauxois | Non-parametric Tests for Recurrent Events under Competing Risks[END_REF], we investigated the case where the parameter of interest is a multivariate trajectory from a set T to R p . We will write that parameter θ 0 (•). The setup is then the space ∞ (T)

p , endowed with the supremum norm :

|| g || T,p := sup t∈T || g(t) || p , g ∈ ∞ (T) p .

The norm || • || T,p is natural when one aims toward establishing confidence regions that are uniform in t ∈ T. Assume that we have a natural estimator θn (•) which can be written

θn (•) = 1 n n i=1 L n (Y i , •), (4.8)
where the t → L n (Y i , t) are observed. The simplest example is when θ 0 (•) = E L 0 (Y 1 , •) for a specified nonrandom map L 0 (•, •), which is not known by the statistician, but which is estimated by L n (•, •).

When T = {t 0 } is a singleton, Hjort et al. [START_REF] Hjort | Extending the scope of empirical likelihood[END_REF] gave very general conditions upon L n to obtain a limit distribution for EL n := max

Sketch of the proof and comments

In this paragraph, I will try to emphasize the aspects of the proof that make it non trivial in regard of the existing literature. Let us go back to the basic ingredients of Owen's results. Consider the setup of Fact 4.2.1, and the expression EL n as defined in (4.7). To simplify the notations, we will consider that µ 0 = E Y 1 = 0. As soon as µ 0 = 0 lies in the convex-hull of the sample (which is asymptotically guaranteed since Y 1 has a nondegenerate covariance matrix). It is possible to express EL n through a Lagrange multiplier λ * = λ * (Y 1 , . . . , Y n ). More precisely, EL n can be expressed as n i=1 np * i , where :

p * i := 1 n 1 1 + λ * • Y i ,
and λ * satisfies

n i=1 1 1 + λ * • Y i Y i = 0.
Then λ * is controlled by making use of the fact that the Y i satisfy the central limit theorem. with α j,n := n j /N .

-The first technical difficulty brought by a multisample setting is the annoying presence of the extra term W j,n (t), itself depending on (p * i,j,n ) i≤n j , in equality (4.11). -The second technical difficulty is to adapt the arguments of Owen to trajectories. Basically :

-Condition (A0 * ) allows the exhibition of the lagrange multipliers λ * (convex hull condition). -The central limit theorem for the Y i is replaced by condition (A1 * ), which is now a convergence of stochastic processes. -The fact that the variance of Y 1 is nondegenerate is replaced by (A2 * ).

-The fact that Y 1 is square integrable is replaced by (A3 * ).

Perspectives

Instead of considering explicit plug-in estimators T (P n 1 ,1 , . . . , P n k ,k ), the next natural step is to investigate M estimators of the general form θn (•) : t → argmin θ∈Θ L P n 1 ,1 , . . . , P n k ,k , θ, t , where L is typically a loss function, and to determine under which regularity conditions upon L it is possible to use the general empirical likelihood principle by unbalancing the empirical weights in the P n j ,j . Another interesting perspective would be to investigate the asymptotic properties of the "most unbalanced empirical measure", namely

P n := n i=1 p * i,n δ Y i ,
where the p * i,n are the maximizing weights of p i f j (Y i ) = P(f j ), j = 1, . . . , p

For example, given a class of functions F, and given (f 1 , . . . , f k ) ∈ F k , is it possible to obtain a limit distribution for the processes

G n := v n P n (•) -P n (•) ,
in ∞ (F), for some normalizing sequence v n ?

4.3 Performances of the discrete associated kernel estimators for the total variation distance

The setup

Let T, B, c be a measured space, where T is countable, B is the σ-algebra of all subsets of T, and c is the counting measure on T. Given an i.i.d. sample (X 1 , . . . , X n ) taking values in T, we are interested in estimating the probability mass function [p.m.f.] f : x → P X = x where X stands for a generic random variable having the common distribution of the X i . The natural estimator of f is the empirical mass function, namely :

f n : x → P n ({x}).
(4.12)

Recently, Kokonendji et al. [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF] and Kokonendji and Abdous [START_REF] Abdous | Consistency and asymptotic normality for discrete associated-kernel estimator[END_REF] introduced the discrete associated kernels density estimator, which can be described as follows : let K := K x,h , x ∈ T, h ≥ 0 be a collection of p.m.f on T. For a bandwidth h ≥ 0, we define :

g n,h (x) := f n,h (x)
T f n,h (u)dc(u)

, where f n,h (x) := 1 n n i=1 K x,h (X i ), x ∈ T.

Several simulation studies [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF] showed promising results, comparing some of those estimators with the traditional "bar plot" f n .

In [START_REF] Abdous | Consistency and asymptotic normality for discrete associated-kernel estimator[END_REF], a first study on the asymptotic properties of such methods showed that their pointwise consistency holds as soon as each K x,h converges, when h → 0, towards the Dirac distribution δ x in the following sense :

∀k = 1, 2, ∀x ∈ T, lim h→0 u k K x,h (u)dc(u) = x k , (4.13)
and ∀x ∈ T, lim sup

h→0 E K x,h (X) 3 < ∞.
Beside the pointwise strong consistency, the authors also established the pointwise asymptotic normality of g n,h (x) (see [START_REF] Abdous | Consistency and asymptotic normality for discrete associated-kernel estimator[END_REF]Theorem 2.4 and 2.5]). With C. Kokonendji, we established asymptotic results for the total variation distance between g n,h and f , namely, the random variable T V g n,h , f := sup Note that, without any further assumptions, it is not even guaranteed that C n,h is a.s. finite. From a statistical point of view, several questions immediately arise :

-For given h ≥ 0, does Y 1,h admit a first/second moment? -Do we have E Y 1,h → 1 as h → 0? -Do we have, for any sequence h n → 0, a convergence of C n,hn to 1, in probability or almost surely? Can we get concentration inequalities?

To answer these questions, we proposed the following sets of assumptions for > 0 :

(HK 0 ) For each x ∈ T, we have K x,h (x) → 1 as h → 0.

(HK 1 ( )) There exist a collection of p.m.f g y (•), y ∈ T and a positive function h such that E h (X) < ∞ and, for each h ∈ [0, 1], (x, y) ∈ T 2 , we have K x,h (y) ≤ h(y)g y (x); (HK 2 )

The function y → sup h∈[0,1] x∈T K x,h (y)dc(x) is bounded on T by a constant C.

When T = Z d , several associated kernels can be represented as where K h , h ≥ 0 is a family of p.m.f on Z d . Our first result is stated as follows.

Theorem 18 (Kokonendji and Varron, 2014, [56]) Let K = K x,h , h ≥ 0 x ∈ T be a family of associated kernels admitting representation (4.15). Then Under (HK 2 ), we have, for all t > 0 and h ∈ [0, 1] :

P C n,h -E(Y 1,h ) ≥ t ≤ exp - n∆ 2 (h) C 2 h C ∆ 2 (h) P C n,h -E(Y 1,h ) ≤ -t ≤ exp - n∆ 2 (h) C 2 h C ∆ 2 (h) ,
where C is mentioned in (HK 2 ), ∆ 2 (h) := Var(Y 1,h ), and h(x) = (x + 1) ln(x + 1) -x, x > 0.

Remark : I has to be noticed that, while condition (HK 0 ) is less stringent than (4.13), this is not the case for conditions (HK 1 ( )) and (HK 2 ). Those conditions may be considered as strong at first glance. However, each associated kernel introduced up to far in the literature does satisfy (HK 1 (2)), sometimes at the price of making light moment assumptions on X.

Results for the total variation distance

Our second result is focused on || g n,h -f || c,1 , note, that, in light of the preceding theorem, and by the inequality

|| g n,h -f || c,1 ≤ C -1 n,h || f n,h -f || c,1 + | C -1 n,h -1 |,
the main object to concentrate our attention on is || f n,h -f || c,1 . We first proved the convergence to 0 of its expectation.

Theorem 19 (Kokonendji and Varron, 2014, [56]) Under (HK 1 (1)) we have : We pursue this subsection with a concentration inequality for || f n,h -f || c,1 . The first one is based on the almost sure boundedness of Y 1,h , while the second is based on the finiteness of a φ α norm, which is closely related to the tail probabilities of Y 1,h . Recall that the φ α norm of a random variable Z is defined as

lim h→0 T E K x,h (
|| Z || φα := inf B > 0, E exp | Z | α B α ≤ 2 , α > 0,
with the implicit convention inf ∅ = ∞.

Theorem 20 (Kokonendji and Varron, 2014, [56]) Under (HK 2 ) we have, for each t ≥ 0 and h ∈ [0, 1].

P || f n,h -f || c,1 ≥ t + E || f n,h -f || c,1 ≤ exp - nt 2 2C 2 , P || f n,h -f || c,1 ≤ -t + E || f n,h -f || c,1 ≤ exp - nt 2 2C 2 .
Now assume that, for some value of α > 0, we have || Y 1,h || φα < ∞ for all h ∈ [0, 1], where

Y i,h := x∈T K x,h (X i ) -E K x,h (X i ) dc(x), i ≥ 1, h ∈ [0, 1].
Then, for any η ∈ [0, 1] and δ > 0, there exists K = K(α, η, δ) such that, for all n ≥ 1, t ≥ 0 and h ∈ [0, 1], we have

P T f n,h (

1. 3 .

 3 CONTRIBUTIONS TO THE STUDY OF LOCAL EMPIRICAL PROCESSES(Hf ) Z 1 admits a Lebesgue density on O for which there exists a version f that is continuous and bounded away from 0 and ∞ on O.

2 .

 2 To go past the unidimensional case. 3. To handle other (continuous) distributions for the (Z i ) i≥1 .

  , A) := inf y∈A d(x, y).

36

 36

2. 1 .

 1 UNIFORM LARGE DEVIATION PRINCIPLES P (h,z) 0 , (recall (2.15)). Moreover, we have, under (Hf ) : lim h→0 sup z∈H sup (g,g )∈G 2 gg dP (h,z) 0 -gg dQ = 0, where Q is the uniform distribution on [-M, M ] d . This convergence of the P (h,z) 0

  This research led to a more general Poissonization technique, which may open new possibilities. To enounce it, I will first introduce some notations. We shall write D := n≥1 D n . Definition 2.2.1 (Truncating maps) Let E 0 be an arbitrary set. A map φ : D → E 0 is said to be truncating if, for any p ≥ 2 and (d 1 , . . . , d p ) ∈ D p we have φ(d 1 , . . . , d p , d p ) = φ(d 1 , . . . , d p ).

  1.1 for X i taking values in an abelian group (D, +), with null element 0 D , and endowed with a norm-like map || • ||, or more generally, a map || • || taking values in R + , for which || 0 D ||= 0, and for which, writing k e := e + . . . + e (k times) and (-k) e := -(k e) for k ∈ N * and 0 e := 0 D , we have

Definition 3 . 1 . 1

 311 Denote by S BV 1 the subset of S 1 of functions f for which 1 0 f 2 (t)dt = 1, and for which f admits a version which has a bounded variation on [0, 1]. Denote by S LIV 1

Definition 3 . 1 . 2 (

 312 Rate function of Poisson type) Let G be a class of functions fulfilling (Support). Define the following rate function on ∞ (G), || • || G .

Corollary 5 . 1

 51 The preceding theorem still holds for a class of function G fulfilling (Support), (Bounded), and which admits finite || • || λ,2 bracketing numbers.

E

  n := e ∈ E, lim n→∞ d(e, E n ) = 0 , and the outer topological limit of (E n ) n≥1 is defined as top lim n→∞ E n := e ∈ E, lim n→∞ d(e, E n ) = 0 .

Definition 3 . 2 . 3 (

 323 Hausdorff distance) In a metric space (E, d), the Hausdorff distance between sets is defined as d Haus (A, B) := max sup ψ∈A d(ψ, B); sup ψ∈B d(ψ, A) .

  15) are due to Einmahl and Mason [42, Proposition A.1].

. 23 )

 23 Then almost surely, for each interval I = [a, b] with 0 ≤ a < b < 1, the following assertion is true :For each closed set F of ∞ ([0, 1]), || • || [0,1] , we have lim n→∞ λ * {z ∈ I, ∆ n (•, h n , z) ∈ F } λ(I) ≤ P(W ∈ F ).

4. 1 A

 1 Donsker and a Glivenko-Cantelli theorem for a class of random measures related to the empirical measure 4.1.1 The framework For p = (p i ) i≥1 ∈ R N and r > 0, we shall write the (possibly infinite) value || p || c,r := i≥1 | p i | r 1/r .

  i , (p i ) i=1,...,n probab. weights,n i=1 np i ≥ u ,

  EL n := max n i=1 np i , (p i ) i=1,...,n probab. weights, n i=1 p i Y i = θ 0 → L exp -χ 2 p . (4.7)

n, i≤n j 1 λ→ P 1 ,WT,p 2 → 1 n

 1121 n i=1 np i , (p i ) i=1,...,n prob. weights, n i=1 p i L n (Y i , t 0 ) = θ 0 (t 0 ) . write the simplex-type sets :S n,u := p = (p i,j ) j≤k,i≤n j , ∀j ≤ k, n j i=1 p i,j = 1, ∀i ≤ n j , p i,j ∈ [0, 1], j p i,j ≥ u , u ∈ [0, 1].Theorem 17 (From Dauxois, Flesch and Varron, 2014[START_REF] Dauxois | Non-parametric Tests for Recurrent Events under Competing Risks[END_REF][START_REF] Varron | Simultaneous confidence bands for some functional plug-in parameters : a computationally feasible approach[END_REF]) On a probability space (Ω, A, P), let W i,j,n (•) n∈N * k , j≤k, i≤n j be a triangular array taking values in ∞ (T)p . For t ∈ T and n ∈ N * k , writeEL n (t) := max k j=1 n j i=1 n j p i,j p ∈ S n,0 , j W i,j,n (t) = 0 .Let V j (•), j ≤ k be deterministic functions on T, whose values are symmetric matrices, and for which0 < inf t∈T, λ∈S(R p ) max j≤k λ • V j (t)λ ≤ sup t∈T, λ∈S(R p ) max j≤k λ • V j (t)λ < ∞.Let (U 1 (•), . . . , U k (•)) be R p -valued stochastic processes. Assume that, for some positive array (a n ) n∈N * k being bounded away from 0, we have, when min{n 1 , . . . , n k } → ∞ :(A0*) inf t∈T, λ∈S(R k ) max j≤k•W i,j,n (t)>0 i,j,n (•) j≤k → L U j (•) j≤k , (A2*) max j≤k a n n j i=1 W i,j,n (•) ⊗2 -V j (•) P 0, (A3*) max j≤k, i≤n j W i,j,n (•) T,p = o P (a -1 n ).Then, when min{n1 , . . . , n k } → ∞ together with → (α 1 , . . . , α k ) ∈ (0, 1] k , we have -2a -1 n log EL n (•) → L U (•) • V -1 (•)U (•),where U (•) := k j=1 α -1 j U j (•) and V (•) = k j=1 α -2 j V j (•).

  A⊂T Ag n,h dc -A f dc = 1 2 || g n,h -f || c,1 . (4.14)4.3.2 The concentration of the normalizing constantIn a first time, we focused on the problem of the normalizing constantC n,h := 1 n n i=1 Y i,h , where Y i,h := T K x,h (X i )dc(x).

K

  x,h (y) := K h (x -y), h ≥ 0, (x, y) ∈ Z d × Z d , (4.[START_REF] Dauxois | Non-parametric Tests for Recurrent Events under Competing Risks[END_REF]

E Y 1 ,h = 1 .

 11 Y 1,h ≡ C n,h ≡ 1 for all h ≥ 0 and n ≥ 1, and (HK 2 ) is satisfied with C = 1. Under (HK 0 ) and (HK 1 ( )), for ∈ N * , we have E Y 1,h < ∞ for all h ∈ [0, 1], and hence E C n,h < ∞ for all n ≥ 1 and h ∈ [0, 1]. Moreover, we have lim h→0 As a consequence, we have, under (HK 0 ) and (HK 1 (2)) : lim h→0 Var(Y 1,h ) = 0, and hence lim h→0 sup n≥1 E C n,h -1 2 = 0.

  X) -f (x) | dc(x) = 0, lim n→∞ sup h∈[0,1] T E f n,h (x) -E K x,h (X) | dc(x) = 0. Moreover, if (HK 1 (2)) satisfied and if T h(y) f (y)dc(y) < ∞, then sup h∈[0,1] T E | f n,h (x) -E K x,h (X) | dc(x) = O(n -1/2 ).

  et Stéphane Chrétien, qui avez par-The central objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The general empirical process . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The local empirical process . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contributions to the study of the general empirical process . . . . . . . . . . . . Contributions to the study of local empirical processes . . . . . . . . . . . . . . Asymptotic results in a spatial setting . . . . . . . . . . . . . . . . . . . 1.3.4 The probabilistic tools that have been developed . . . . . . . . . . . . . . 1.4 Inventory of publications and prepublications . . . . . . . . . . . . . . . . . . . Uniform large deviation principles . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Large deviations in spaces having the bounded approximation property . 2.1.3 A ULDP for the local empirical process . . . . . . . . . . . . . . . . . . . 2.1.4 Asymptotic independence of two local empirical processes at very different scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Some words on bracketing . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poissonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The negligible effect of sample size randomization for large deviation events 2.2.2 A general Poissonization tool for i.i.d. sums in a measurable semigroup . 2.2.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Contributions to the study of local empirical processes 3.1 Improvements of existing functional limit laws . . . . . . . . . . . . . . . . . . . 3.1.1 Some preliminary words on the usual tools used in functional limit laws . 3.1.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Clustering rates and Chung-Mogulskii functional limit laws for the local uniform empirical process . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Nonstandard functional limit laws . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Uniform-in-bandwidth functional limit laws . . . . . . . . . . . . . . . . . . . . . 3.2.1 Asymptotic independence in the local standard functional limit law . . . .2.7 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Spatial-type limit laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 A Glivenko-Cantelli theorem . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 A Donsker theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Sketch of the proof and comments . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Empirical processes theory for empirical likelihood . . . . . . . . . . . . . . . . . 4.2.1 The empirical likelihood principle . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Estimations of linear, trajectory-valued functionals . . . . . . . . . . . .

	Contents Introduction 1 Introduction 1.1 2 Uniform large deviations and Poissonization tools for local empirical pro-cesses Introduction 2.1 Chapter 1 1.1 The central objects

1.2.1 The uniform entropy condition . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Contributions to empirical likelihood theory . . . . . . . . . . . . . . . . 1.2.3 Contributions to associated discrete kernel estimation . . . . . . . . . . . 1.3 1.3.1 Refinements of existing functional limit laws . . . . . . . . . . . . . . . . 1.3.2 The impact of the in-bandwidth uniformity to the usual functional limit laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 1.4.1 Short notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Full length articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Submitted articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Articles in preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 2.2 3.2.2 In bandwidth uniformity in the local standard functional limit law . . . . 3.2.3 Asymptotic independence in the global standard functional limit law . . 3.2.4 In bandwidth uniformity for the global standard limit law . . . . . . . . 3.2.5 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 A nonfunctional uniform-in-bandwidth limit law for an object related to nonparametric regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3.1 A spatial Donsker theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Spatial functional limit laws . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Contributions to the study of the general empirical process, and its applications to statistics 4.1 A Donsker and a Glivenko-Cantelli theorem for a class of random measures related to the empirical measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 4.2.3 Estimation of nonlinear functionals . . . . . . . . . . . . . . . . . . . . . 4.2.4 The underlying probabilistic result . . . . . . . . . . . . . . . . . . . . .

  1.3. CONTRIBUTIONS TO THE STUDY OF LOCAL EMPIRICAL PROCESSESsums of i.i.d. random variables in a measurable semigroup, proving that

  Mason could be extended to obtain uniform large deviation principles of the form

	lim n→∞	sup z∈Hn,
		h∈[hn,hn]

  For each p ≥ 1 and t = (t 1 , . . . , t p ) ∈ T p , the sequence of collections Π t (W n,ρ ) n≥1, ρ∈Hn is Borel measurable, and satisfies the ULDP in R p , for (v n,ρ ) n≥1, ρ∈Hn and for a rate function J t .

	T , p ≥ 1 and t = (t 1 , . . . , t p )
	Π t (ψ) := ψ(t 1 ), . . . , ψ(t p ) .	(2.2)
	Proposition 2.1.1 (Varron, derived from Arcones, 2004) Take (E, O) as the Banach space
	( ∞ (T ), || • || T ). Assume that
	1. (Finite dimensional ULDP)

  ). Now let us go back to our initial ULDP problem. Straightforward analysis (see, e.g., [5, Lemma 4.1]) shows that (recall (2.2)) :

  and n ≥ 1, inserting (2.27) in Fact 2.1.2 (applied to G(h, z )) leads to :

				m
	P	max m=1,...,n	sup g∈G	i=1

  .32) Now (2.32) is proved by noting first that, by Lemma 2.1.1, we have, for fixed (g, g ) ∈ G 2 :

  g Borel and bounded.(2.33) When each η n is Poisson with expectation n and independent of (Z i ) i≥1 , the Π ηn (•, h, z) are called Poissonized versions of the local empirical process (or local Kac processes), and shall be written Π

n (•, h, z) for short. The main feature of Π n (•, h, z) is that those processes inherit of a crucial property of Poisson random measures, which, in terms of empirical measures, can be written as follows.

Proposition 2.2.1 If (G 1 , . . . , G p ) are classes of functions respectively vanishing out of disjoint subsets J 1 , . . . , J p , then the family of processes

  This general framework encompasses that of Proposition 2.2.4, by choosing φ(s 1 , . . . , s p ) = s p , p ≥ 1, (s 1 , . . . , s p ) ∈ D p .

It also encompasses maxima of partial sums, by choosing, when D is a normed space :

φ(s 1 , . . . , s p ) = max j=1,...p || s j ||, p ≥ 1, (s 1 , . . . , s p ) ∈ D p .

Proof : Write p B := P(X 1 ∈ B). By an application of Stirling's formula, and since (1-p B ) > 0, we have

  The counterpart of Step (Expectation) is simple : for each z ∈ [0, b[, and for h n ≤ [1-b], the law of ∆ n (•, h n , z) is exactly the law of ∆ n (•, h n , 0), hence, writing, for any bounded Lipschitz function φ on ∞ ([0, 1]), || • || [0,1] :

  by independence of the increments -The counterpart of (Interpolation) is the use of the usual blocking arguments, which hold between consecutive terms of geometrical subsequences n k ∼ (1 + γ) k . Combining those arguments with the preceding variance calculus concludes the proof, but under the technical condition that h n k is summable in k for any geometric subsequences n k . This explains why I had to impose condition lim n→∞ log(1/h n )/ log log(n) > 1 in (3.23), which excludes sequences h n tending too slowly to 0. I do not know if this condition can be bypassed or if some new phenomenon arises when, e.g., lim n→∞ log(1/h n )/ log log(n) < 1.

  The Poissonization technique of Giné et. al (Proposition 2.2.4) adapts with very few efforts to the objects T n (•, h, z) (for more details, see [101, Proposition 2]).By assumption (Support), we have Π n (•, h, z) ⊥ ⊥ Π n (•, h, z ) as soon as || z -z || d ≥ 2M h. Hence, the (V ariance) argument can be easily generalized. The assumption that µ admits a bounded density ensures that the measure of the diagonal µ ⊗2

  where both the probability weights (β i ) i≥1 and the locations Y i are random. Since the empirical measure P n , as well as its various bootstrapped versions (see, e.g.,[START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF] Chapter 3.6]) belong to that class of objects, I investigated how far some usual techniques in empirical processes theory could be used in the study of that wider class of random measures. I found that, beside some non surprising envelope conditions, condition (U nif. entropy), with the presence of the envelope in the integrals, is robust enough to obtain Donsker and Glivenko-Cantelli theorems, when the Y i are conditionally i.i.d. given the weights. Those results are stated in §4.1.-The empirical likelihood technique, initiated by Thomas and Grunkemeier[START_REF] Thomas | Confidence interval estimation of survival probabilities for censored data[END_REF] is an alternative to Gaussian pivot or bootstrap procedures for the construction of confidence regions. Roughly speaking, it consists in building confidence regions around an estimator by slightly unbalancing the empirical weights 1/n in the definition of the estimator. With J-Y Dauxois and A. Flesch, we worked on using the empirical processes theory to prove that a general class of empirical likelihood procedures is consistent. Those results are stated in §4.2. -The discrete associated kernel estimator is a general class of nonparametric estimators of the probability mass function of a discrete random variable. Despite the "kernel" terminology, such estimators are not related to the local empirical measure, due to the discrete nature of the data. With C. Kokonendji, we investigated the asymptotic properties of these estimators under the total variation distance. Those results are stated in §4.3.

  || • || P n,β n (ω),2 , for almost all ω. Assertion (4.6) is a consequence of the fact that W Yn,β n is, conditionally to β n , Gaussian with intrinsic semi metric || • || P n,β n ,2 . Assertion (4.5) is a consequence, through symmetrization, of the fact that, conditionally to β n and Y n , the process

i≥1 i β i,n δ Y i,n (•)

is subgaussian with intrinsic semimetric P (β n , Y n ) (recall (4.2)). Then condition (U nif. entropy) plays its role.

  Now, if we try to use the same representation for t → EL n (t), we obtain EL n (t) = (t) j≤k, i≤n j ∈ S n,0 , and (γ * j ) j=1,...,k ∈ R k . Summing (4.10) in i for fixed j entailsγ * j (t) + N λ * n (t) • W j,n(t) = n j , with W j,n (t) :=

	k	n j
		n j p * i,j (t), with
	j=1	i=1
		p * i,j,n (t)γ * j (t) + N p * i,j,n (t)λ *
		k	n j
			p * i,j,n (t)W i,j,n (t) = 0,
		j=1	i=1
		p * i,j,n n j
					p * i,j,n (t)W i,j,n (t).
					i=1
	This leads to the expression
			p * i,j,n (t) :=	1 n (t) • (W i,j,n (t) -W j,n (t)) n j + N λ *
			=	1 n j	1 + α -1 j,n λ	1

n (t) • W i,j,n (t) = 1, j ≤ k, i ≤ n j , (4.10) * n (t) • (W i,j,n (t) -W j,n (t)) ,

(4.11)

  As a consequence, for any sequence h n → 0, we have || f n,hn -f || c,1 → 0 almost completely.

	≤ exp -	nt 2 8(1 + δ)σ 2 (h)	+ 3 exp -	n α t α
	≤ exp -	nt 2 8(1 + δ)σ 2 (h)	+ 3 exp -	n α t α

x) -E f n,h (x) dc(x) ≥ t + (1 + η)E T f n,h (x) -E f n,h (x) dc(x) K || Y 1,h || φα log(n) 1/α

, and

P T f n,h (x) -E f n,h (x) dc(x) ≤ -t + (1 -η)E T f n,h (x) -E f n,h (x) dc(x) K || Y 1,h || φα log(n) 1/α

, where

σ 2 (h) := sup A⊂T Var A K x,h

(

X)dc(x) .

2.1. UNIFORM LARGE DEVIATION PRINCIPLES

Assume in addition that

Then, almost surely, Θ global,⊗2 n (h n , h n , H) n≥2 converges to S ⊗2 G (α) for the Hausdorff distance.

Sketch of the proof and comments

The structure of the limit set S ⊗2 G,1 (α) depends on α := lim n→∞ log(1/h n )/ log(1/h n ). That parameter measures how different the rates of convergence to 0 of h n and h n are (typically, h n and h n are two different powers of 1/n). Let us consider the two extreme cases :

-When α gets close to 1, this means that those rates of convergence tend to be similar (but note that h n is always o(h n ), whatever the value of α < 1). In that case, the limit set S ⊗2 G (α) is close to the limit set exhibited in Theorem 6. Theorem 8 then has the following rough interpretation : if, for some z ∈ H, the process -When α gets close to 0, this means that h n and h n have very different rates of convergence to zero. In that case, the limit set S ⊗2 G,1 (α) can have elements (ψ 1 , ψ 2 ) with ψ 1 simultaneously having their respective stochastic costs close to 1. Hence, the compensation phenomenon softens : it may happens (in fact, it happens almost surely for all large n) that, for some z ∈ H, both the processes (2f (z)nh

simultaneously hit (neighborhoods of) functions having respective stochastic costs 1 and 1 -α.

I will now give a heuristic argument to understand why this strange phenomenon occurs :

1. As showed in the heuristics of the proof of Fact 3.2.1, H contains, for each n, a number M n ∼ λ(H)h -d n of points z n, (distant one from each other of at least h n ), which define "mutually independent processes" (2f (z n, )nh

). Given ψ 2 fulfilling J G (ψ 2 ) ≤ 1, the probability that at least one of these processes hits ψ 2 is high enough because "we run M n independent trials". 2. But, for a given small value δ > 0, since

of points z n, ,q , q ≤ m n , which are distant from each other of at least h n (hence defining "independent" processes (2f (z n, ,q )nh d n log(1/h d n )) -1/2 T n (•, h n , z n, ,q ) but which are still at distance at most δh n of their corresponding point z n, . That proximity with respect to the scale h n allows to approximate all the (2f (z n, ,q

Under the above mentioned assumptions, Einmahl and Mason have proved the following theorem, λ denoting the Lebesgue measure.

Theorem 10 (Einmahl, Mason, 2000) Assume that the law of (Y, Z) fulfills (Hf ) for some product of open set

, (HC) and (HK1) -(HK3) are satisfied. Let h n be a sequence fulfilling (HV ).

Then, given a compact H ⊂ O 2 with nonempty interior we have, almost surely :

Their proof uses the same discretization techniques as in Theorem

In fact, our result in [START_REF] Van Keilegom | Uniform in bandwidth exact rates for a class of kernel estimators[END_REF] has a weaker form that assertion (3.21), but is can be extended with no major technical difficulty.

Since their initial work in [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF], Einmahl and Mason (sometimes in collaboration with Dony) did prove in-bandwidth-uniform results for objects such as V n (g, h, z) (see [START_REF] Dony | Weighted uniform consistency of kernel density estimators with general bandwidth sequences[END_REF][START_REF] Dony | Uniform in bandwidth consistency of kernel regression estimators at a fixed point[END_REF][START_REF] Dony | Uniform in bandwidth consistency of local polynomial regression function estimators[END_REF][START_REF] Dony | Uniform in bandwidth consistency of conditional U -statistics[END_REF][START_REF] Dony | Uniform in bandwidth consistency of kernel estimators of the tail index[END_REF]).

Our contribution with regard of these results is that we could make explicit the limits and normalizations, at the price of slightly strengthening conditions upon h n an h n .

Perspectives

A first perspective of improvement will be to relax the (V C subgraph) assumption to a condition of the type (U nif. entropy). At the light of the improvement of Fact 3.2.1, I am pretty certain that such an extension is reachable. A second perspective of improvement would be to relax the assumption that the whole vector

EMPIRICAL PROCESSES THEORY FOR EMPIRICAL LIKELIHOOD

Following their ideas, we could derive a convergence for processes (in ∞ (T) p ) of the type

toward a stochastic process which is the square Euclidian norm of an R p -valued gaussian process on T. Such a limit law permits to construct asymptotic confidence bands for θ 0 (•).

Original motivation from lifetime data analysis

The initial motivation of our work came from a particular inference problem in survival analysis : the inference of the mean number of different competing recurrent events. That framework was former introduced by Dauxois and Sencey [START_REF] Dauxois | Non-parametric Tests for Recurrent Events under Competing Risks[END_REF], and can be described as follows : We observe i.i.d. replicates of the following random phenomenon :

-Consider a finite collection of counting processes on a time interval, N j (•)) j=1,...,k .

Those coutning processes represent k competing, recurrent events (for example, different type of nosocomial infections that can be contracted by hospitalized patients). Let N j (t) denote the number of total events of type j up to time t. -Let D be a terminal event and C an independent censorship. We make no assumption on the dependence structure between the counting processes and D. We suppose that Let us denote by S the survival function of D, that is S(t) = P(D > t). Write also :

The observed data are i.i.d. replicates (N i,j (•), X i , δ i ) j≤k of (N j (•), X, δ) j≤k , where the processes are observed on [0, τ ], and τ is a fixed constant chosen so that P(C > τ )P(X > τ ) > 0. In order to build confidence regions for the mean functions t → µ j (t) = E(N j (t)), we use the estimators µ j (t) from Dauxois and Sencey [START_REF] Dauxois | Non-parametric Tests for Recurrent Events under Competing Risks[END_REF] :

where S(•) is the Kaplan-Meier estimator of S(•), the survival function of D (see, for example, [START_REF] Andersen | Statistical models based on counting processes[END_REF] for a definition of S(•)). Such an estimator can typically be described as (4.8).

Estimation of nonlinear functionals

Among the possible generalizations of Owen's results, one particular approach is to consider parameters of interests that can be written θ 0 := T (P 0 ), where P 0 is the common law of the observed sample (Y 1 , . . . , Y n ), and where T is a specified map. In that case, the idea is to build the following type of confidence region.

and determine their asymptotic covering probabilities, under the smallest possible assumptions upon the functional T .

The first results establishing an ad-hoc convergence in distribution such as (4.7) were already present in Owen [START_REF] Owen | Empirical likelihood ratio confidence intervals for a single functional[END_REF], but they were limited to real-valued data, together with T being Frechet differentiable in D(R), || • || R , where D(R) stand for the space of bounded CADLAG functions on R. Frechet differentiability is a rather strong assumption, which is rarely satisfied in practice.

The next major step in that direction was taken by Bertail [START_REF] Bertail | Empirical likelihood in some semiparametric models[END_REF], where he showed that, if a class of functions F satisfies (U nif. entropy) and admits a square integrable envelope then a limit result such as (4.7) holds for functional T that are Hadamard differentiable with respect to || • || F , tangentially to C(F, || • || P 0 ,2 ), the latter being the space of || • || P 0 ,2 continuous functions on F.

Following an initial work in A. Flesch's PhD thesis, I worked on extending Bertail's theorem under three directions :

-I considered the more general case where θ 0 is a trajectory (as in the preceding subsection). -In practice, computing R n,u is computationally too intensive (an exponential n complexity). Following the original idea of A. Flesch, I investigated the consistency of other confidence regions R LP n,u , which involve a local linearization procedure of T around P n , and which makes the computation times reasonable.

-I considered the case where the estimator is built on several (say k) mutually independent samples. More precisely, θ 0 = T (P 0,1 , . . . , P 0,k ) is estimated by T (P n 1 ,1 , . . . , P n k ,k ), using the corresponding empirical measures. Note that some steps were already taken in the multisample direction, for estimating differences of means or proportions ([53, Theorem 1], [78, statement following (2.6)], and also [START_REF] Liu | Empirical likelihood for the two-sample mean problem[END_REF]Theorem 1] in the multivariate case), but the corresponding proofs were never entirely written.

The underlying probabilistic result

The two preceding paragraphs are voluntarily evasive, because introducing all the ad-hoc notations and assumptions would take a lot of space for an already quite long manuscript. It is however possible to point out the key probabilistic result which is behind those two contributions, without much notations. We will write n := (n 1 , . . . , n k ) ∈ N * k , and we will