Je Remercie

Clément Dombry

Je Remercie Également

Jérôme Dedecker

Davide Giraudo

Paul-Marie Samson

Viktor Mirella Gauthier

Cecile Benjamin

Keywords: Champs aléatoires, Processus stationnaires, Ortho-martingales, Théorème limite central, Principe d'invariance faible, Processus de Weiner, Approximation par ortho-martingales, Décomposition en cobord, Critères projectifs, Espaces de Banach, Distribution empirique, Distance L p, Distance de Wasserstein 60F17, 60G60, 60F25, Random elds, Stationary processes, Ortho-martingales, Central limit theorem, Weak invariance principle, Weiner process, Ortho-martingales approximation, Coboundary decomposition, Projective criteria, Banach spaces, Empirical distribution, L p distance, Wasserstein distance AMS classication: 60F17, 60G60, 60F25,

Je commence par remercier ma directrice de thèse Florence Merlevède, et mon directeur de thèse Dalibor Volný pour leurs accompagnements durant ces 3 ans (8 ans pour Dalibor). Tous vos conseils, votre aide, les références pointues, et toutes les relectures attentives des articles envoyés sont essentiels pour ma thèse. Je suis extrêmement reconnaissant d'avoir bénécié de votre expertise, et je vous remercie pour tout le temps que vous avez consacré à mon encadrement. En plus de toutes les connaissances mathématiques que vous m'avez transmises, je tiens à remercier Florence pour sa grande patience. Elle m'a aidé à corriger toutes les fautes d'orthographe, mot pour mot, que ce soit dans les articles ou non. Je pense que personne ne peut demander plus. C'est Dalibor qui m'a initié à la recherche avec un super mémoire de Master sur les théorèmes limites. L'article de Kipnis et Varadhan 1986 que tu m'avais montré aura toujours une place spéciale dans mes souvenirs. Je tiens à te remercier pour m'avoir donné l'envie de voir plus et le courage d'aller plus loin.

Chapter 1 Introduction

Dans cette thèse nous nous intéressons à des théorèmes centraux limites pour des champs aléatoires non nécessairement indépendants. Nous travaillerons tout au long de cette thèse avec des champs aléatoires strictement stationnaires. Ce sont des variables aléatoires indexées par Z d telles que les lois ni-dimensionnelles sont invariantes par translation du temps. Pour dénir les champs aléatoires strictement stationnaires, on introduit naturellement les Z d -actions dans un espace probabilisé (Ω, A, P). On note les éléments de Z d par i := (i 1 , . . . , i d ) ∈ Z d . On dénit maintenant le groupe des Z d -actions (T i ) i∈Z d dans un espace probabilisé (Ω, A, P). Ces Z d -actions sont générées par {T e 1 , . . . , T e d } qui sont des transformations bimesurables préservant la mesure P, où pour tout 1 ≤ i ≤ d, e i est un vecteur de Z d avec 1 à i-ième place et 0 ailleurs. Soit alors (X • T i ) i∈Z d un champ aléatoire strictement stationnaire dénie sur (Ω, A, P). On notera X • T k par X k . Une des problématiques de cette thèse sera d'établir des conditions susantes pour obtenir la convergence asymptotique suivante :

1 |n| S n (X) ⇒ σN (0, 1) , où S n (X) := n 1 k 1 =1 • • • n d k d =1 X • T k et |n| = n 1 × • • • × n d .
Ici ⇒ désigne la convergence en loi, et N (0, 1) la loi normale centrée réduite. Dans le cas d = 1, et si les X i sont indépendantes, et identiquement distribuées (i.i.d.) et dans L 2 , le théorème central limite (TCL) assure que n -1/2 S n (X) converge en loi vers une loi normale. On dira également que le TCL a lieu pour la suite (X • T i ) i∈Z . Dans le cadre indépendant, le théorème peut se généraliser aisément aux champs aléatoires avec d ≥ 2. On peut également obtenir la version fonctionnelle de ce résultat, aussi appelé principe d'invariance faible (WIP). Plus précisément, on cherche à établir le comportement asymptotique suivant pour le processus des sommes partielles :

1 √ n S [nt] (X) t∈[0,1] ⇒ {σB t } t∈[0,1] ,
où (B t ) t∈[0,1] est un mouvement brownien. Dans le cadre i.i.d, ce résultat est dû à Donsker [START_REF] Donsker | An invariance principle for certain probability limit theorems[END_REF]1951]. Pour de nombreux modèles, l'hypothèse d'indépendance n'est pas satisfaite et étendre ces résultats dans le cadre des suites ou des champs dépendants a donc de multiples applications. Dans la suite, nous commencerons par rappeler quelques résultats concernant ces problématiques.

Théorèmes limites pour les suites de variables aléatoires dépendantes Soit (Ω, A, P) un espace probabilisé. Comme dans la section précédente, on dénit les suites strictement stationnaires à l'aide d'une variable aléatoire dans (Ω, A, P) et d'une transformation T bijective, bimesurable qui préserve la mesure P. Ainsi on considère la suite (X k ) k∈Z dénie par X k = X • T k où X est une variable aléatoire dans (Ω, A, P). On considèrera aussi une ltration (F k ) k∈Z stationnaire dénie par

F k = T -k (F 0
) où F 0 est une sous tribu de A telle que F 0 ⊂ T -1 (F 0 ). Les suites de diérences de martingale étant un exemple de suites dépendantes, commençons par en rappeler la dénition.

Dénition (A1). Soit d ∈ L 1 (P), on dénit la suite strictement stationnaire (d k := d • T k ) k∈Z . On dit que (d k ) k∈Z est une suite de diérences de martingale pour la ltration (F k ) k∈Z si d est F 0 -mesurable et E (d|F -1 ) = 0 p.s. . D'autre part, M n := n k=1 d k est une martingale pour la ltration (F k ) k∈Z .

Nous sommes ainsi en mesure d'énoncer le théorème central limite pour les martingales de carré intégrable à accroissements strictement stationnaires. Ce théorème est dû à Ibragimov [START_REF] Ibragimov | A central limit theorem for a class of dependent random variables[END_REF]1963] et Billingsley [START_REF] Billingsley | The Lindeberg-Lévy theorem for martingales[END_REF]1961] (le résultat a été obtenu de façon indépendante).

Théorème (B1). Soit (d k := d • T k ) k∈Z une suite strictement stationnaire de différences de martingale pour la ltration (F k ) k∈Z . Si d est de carré intégrable, et T est ergodique alors

1 √ n n k=1 d • T k ⇒ N (0, σ 2 ) , où σ 2 = ∥d 0 ∥ 2 2 .
Dans [START_REF] Gordin | The central limit theorem for stationary processes[END_REF]1969], Gordin fut l'un des premiers à exploiter ce théorème dans un contexte de dépendance au sens large, en utilisant notamment la décomposition en cobord. Cette décomposition consiste à écrire :

X = d + θ -θ • T , (1)
où (d • T n ) n∈Z est une suite de diérences de martingale dans L 1 , et ([θ -θ • T ] •

T n ) n∈Z est le cobord. On en déduit alors que

S n = n k=1 d k + (θ -θ n ) . (2)
On remarque que si la suite (d n ) n∈Z est ergodique dans L 2 (P) alors elle satisfait le TCL pour les suites des diérences de martingale. Ainsi, si le cobord renormalisé est négligeable pour la convergence en loi, alors le TCL pour (X n ) n∈Z se déduira de celui de (d n ) n∈Z . Gordin a ensuite proposé le TCL suivant à l'aide de cette décomposition en cobord.

Théorème (B2). Soit (X k ) k≥0 une suite strictement stationnaire et ergodique dans L 2 (P), adaptée à la ltration (F i ) i≥0 . Suppose que

k≥1 ∥E (X k |F 0 )∥ 2 < ∞ . (3)
Alors n -1/2 S n (X) converge en loi vers une loi normale centrée de variance σ 2 = ∥d 0 ∥ 2 2 := k∈Z Cov(X 0 , X k ).

Heyde [START_REF] Heyde | On the central limit theorem and iterated logarithm law for stationary processes[END_REF]1975] a ensuite montré que le WIP est satisfait dès que l'on a

k≥1 E (X k |F 0 ) converge dans L 2 . (4)
D'autre part, Volný [START_REF] Volný | Approximating martingales and the central limit theorem for strictly stationary processes[END_REF]1993] a montré que (4) est une condition nécessaire et susante pour obtenir la décomposition (1), avec d et θ dans L 2 . Une question naturelle est donc de savoir si le TCL reste vrai, si on relaxe (3) demandant seulement la convergence L 1 . Dans [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF]1973] Gordin a donné une réponse positive à cette question et a établi le TCL suivant :

Théorème (B3) [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF]. Soit (X k ) k∈Z une suite strictement stationnaire et ergodique dans L 1 (P) adapté à la ltration (F i ) i∈Z . Supposons que i≥0 E(X i |F 0 ) converge dans L 1 (P)

et

lim inf n→∞ E(|S n (X)|) √ n < ∞ . (6)
Alors n -1/2 S n (X) converge en loi vers une loi normale centrée.

C.G. Esseen et S. Janson [START_REF] Esseen | On moment conditions for normed sums of independent variables and martingale dierences[END_REF]1985] ont ensuite donné une démonstration détaillée de ce Théorème en utilisant l'inégalité de Burkholder suivante Lemme (C1). Soit (d k ) k∈Z une suite de diérences de martingale dans L 1 (P). Il existe une constante universelle C tel que pour tout n ≥ 1 et λ > 0, on a

P   n k=1 d 2 k 1/2 > λ   ≤ C λ E (|S n (d)|) .
En eet, la condition [START_REF] Heyde | On the central limit theorem and iterated logarithm law for stationary processes[END_REF] entraine la décomposition en cobord de (X k ) k≥0 dans L 1 (P) (voir [START_REF] Volný | Approximating martingales and the central limit theorem for strictly stationary processes[END_REF]1993]). Il existe donc (d k ) k≥0 une suite de diérences de martingale dans L 1 (P) telle que S n (X) = n k=1 d k -Z n avec sup n≥1 ∥Z n ∥ 1 < ∞. Ainsi, si (d k ) k≥0 est dans L 2 (P), elle satisfait le TCL pour les diérences de martingale, et entrainera le TCL pour la suite (X k ) k≥0 . Supposons par simplicité que n -1/2 E (|S n (X)|) ≤ A < ∞. Comme sup n≥1 ∥Z n ∥ 1 < ∞, on en déduit que n -1/2 E (|S n (d)|) ≤ B < ∞. Par le lemme (C1), pour λ assez grand, on a :

P   1 n n k=1 d 2 k 1/2 ≥ λ   ≤ C λ E |S n (d)| √ n ≤ CB λ < 1 . (7)
Or, par le théorème ergodique

1 n n k=1 d 2 k -→ E d 2 0 presque sûrement quand n → ∞ .
Si E (d 2 0 ) = ∞, alors n -1 n k=1 d 2 k -→ ∞ presque sûrement, et la premier partie de l'inégalité [START_REF] Cuny | On martingale approximations and the quenched weak invariance principle[END_REF] converge vers 1 quand n converge vers inni, d'où la contradiction. Cependant cette démonstration ne peut pas être généralisée pour les champs aléatoires, car l'inégalité de Burkholder n'a pas d'équivalent pour les ortho-martingales qui peuvent être vues comme la version multi-index des martingales. Le chapitre 2 de cette thèse est consacré à l'étude de l'extension du Théorème (B3) aux champs aléatoires. An d'exposer les résultats obtenus, introduisons quelques notations.

Notation pour des champs aléatoires dans Z 2 . En complément des notations précédentes, on introduit S une transformation bijective, bimesurable qui préserve la mesure P. On dénit ainsi un champ aléatoire strictement stationnaire de la façon suivante : Soit (X k,ℓ ) k,ℓ∈Z le champ aléatoire strictement stationnaire déni sur (Ω, A, P) par X k,ℓ := X • T k • S ℓ , avec X une v.a. dénie sur (Ω, A, P). Soit F 0,0 une sous tribu de A. Pour tout i, j ∈ Z, on dénit F i,j = T -i S -j F 0,0 . On dit que (F i,j ) i,j∈Z est une ltration, si pour tout j ∈ Z xé, on a F i,j ⊂ F k,j pour tout i ≤ k et de même pour indice j. Introduisons également la notion de ltration commutante : Dénition (A2). On dit que la ltration (F i,j ) (i,j)∈Z 2 est complètement commutante si ∀i, j, h, k ∈ Z, et pour tout v.a. X, on a

E [E [X|F i,j ] |F h,k ] = E [E [X|F h,k ] |F i,j ]
= E [X|F i∧h,j∧k ] p.s. , avec i ∧ j = min(i, j).

Nous pouvons maintenant donner la dénition des ortho-martingales. Dénition (A3). Soit (d i,j ) (i,j)∈Z 2 un champ aléatoire intégrable adapté à une ltration complètement commutante (F i,j ) (i,j)∈Z 2 . On dit que (d i,j ) (i,j)∈Z 2 est un champ de diérences d'ortho-martingale pour la ltration (F i,j ) (i,j)∈Z 2 , si pour tout i, j, k, ℓ ∈ Z, tel que k < i, ou ℓ < j on a :

E (d i,j |F k,ℓ ) = 0 p.s.
De plus, M n 1 ,n 2 := n 1 k= n 2 ℓ=1 d k,ℓ est une ortho-martingale pour la ltration (F k,ℓ ) k,ℓ∈Z .

Soit (X i,j ) (i,j)∈Z 2 un champ aléatoire strictement stationnaire adapté à la ltration complètement commutante (F i,j ) (i,j)∈Z 2 dénie sur (Ω, A, P). On peut établir une décomposition ortho-martingale-cobord. Elle est due à Gordin [START_REF] Gordin | Martingale-coboundary representation for a class of random elds[END_REF]2009] (voir aussi [START_REF] El Machkouri | Orthomartingale-coboundary decomposition for stationary random elds[END_REF]2016] pour une application au TCL). Pour dénir correctement la décomposition, introduisons les notations supplémentaires :

F i,-∞ = T -i σ j∈Z S -j F 0,0 et F -∞,j = S -j σ i∈Z T -i F 0,0 .
La décomposition ortho-martingale-cobord doit satisfaire : [START_REF] Gordin | Martingale-coboundary representation for a class of random elds[END_REF] où (d i,j := d • T i • S j ) (i,j)∈Z 2 est une diérence d'ortho-martingales pour la ltration (F i,j ) (i,j)∈Z 2 . Pour tout j ∈ Z xé, ([g 2 -g 2 • S] • T i • S j ) i∈Z est une diérence de martingale pour la ltration (F i,∞ ) i∈Z , et pour tout i ∈ Z xé, ([g 2 -g 2 • S] • T i • S j ) i∈Z renormalisé est négligeable pour la convergence en loi. Pour tout i ∈ Z xé, ([g 1 -g 1 • T ] • T i • S j ) j∈Z est une martingale pour la ltration (F ∞,j ) j∈Z et pour tout j ∈ Z xé, ([g 1 -g 1 • T ] • T i • S j ) i∈Z renormalisé est négligeable pour la convergence en loi. Ces deux champs sont appelés les cobord mixtes. Le champ 2 est le cobord qui, renormalisé, est négligeable pour la convergence en loi.

X = d + (g 1 -g 1 • T ) + (g 2 -g 2 • S) + (g 3 -g 3 • T -g 3 • S + g 3 • T • S) ,
([g 3 -g 3 • T -g 3 • S + g 3 • T • S] • T i • S j ) (i,j)∈Z
Tout comme dans le cadre des suites, il existe une condition nécessaire et susante pour obtenir la décomposition [START_REF] Gordin | Martingale-coboundary representation for a class of random elds[END_REF]. En particulier Volný [START_REF] Volný | Martingale-coboundary representation for stationary random elds[END_REF]2018] a montré que i,j≥0 E(X i,j |F 0,0 ) converge in L 1 (P) [START_REF] Dede | An empirical central limit theorem in L1 for stationary sequences[END_REF] est une condition susante pour obtenir la décomposition (8) (elle est également nécessaire dans le cadre régulier (voir dénition 2.1.2)). Dans le chapitre 2, nous établirons le résultat suivant :

Théorème (B4). Soit (X k,ℓ ) k,ℓ≥0 un champ aléatoire centré et ergodique dans L 1 (P), et adapté à la ltration complètement commutante (F i,j ) i,j≥0 . On pose S n 1 ,n 2 (X) = n 1 -1 i=0 n 2 -1 j=0 X i,j . Supposons [START_REF] Dede | An empirical central limit theorem in L1 for stationary sequences[END_REF] et que les conditions suivantes sont satisfaites :

lim inf n→∞ 1 √ n n-1 i=0 X i,0 1 < ∞ , lim inf N →∞ 1 √ N N -1 j=0 X 0,j 1 < ∞ (10)
et lim inf

n 1 →∞ lim inf n 2 →∞ E(|S n 1 ,n 2 (X)|) √ n 1 n 2 < ∞ , lim inf n 2 →∞ lim inf n 1 →∞ E(|S n 1 ,n 2 (X)|) √ n 1 n 2 < ∞ . (11)
Alors, les v.a. d, g 1 , g 2 et g 3 dénies dans [START_REF] Gordin | Martingale-coboundary representation for a class of random elds[END_REF] sont dans L 2 (P).

Cependant contrairement au cadre des suites (d = 1), le résultat précédent ne garantit pas le TCL. Cela vient du fait que les cobord mixtes renormalisés ne sont pas forcément négligeables dans les deux directions (voir le contre-exemple 3.1.4). Ainsi une condition supplémentaire est nécessaire pour que les cobords mixtes renormalisés soient négligeables. C'est l'objet du résultat suivant : Théorème (B5). Sous les conditions de théorème (B4), et si

lim min(n 1 ,n 2 )→∞ E(|S n 1 ,n 2 (X)|) √ n 1 n 2 existe , (12)
alors (n 1 n 2 ) -1/2 S n 1 ,n 2 (f ) converge en loi vers une variable gaussienne centrée (qui peut être dégénérée).

D'autres types de conditions projectives peuvent également être considérés et induisent une approximation par une ortho-martingale. Ces conditions n'entrainent pas une décomposition en cobord mais assurent la négligeabilité des "restes renormalisés". En particulier, dans le cadre régulier (voir dénition 2.1.2), la condition de Hannan [START_REF] Hannan | Central limit theorems for time series regression, Z. Wahrscheinlichkeitstheor und verw[END_REF]1973] :

k≥1 ∥P 0 (X k )∥ 2 < ∞ (13) où P i (X k ) := E (X k |F i )-E (X k |F i-1
), est susante pour assurer que la suite (X n ) n∈Z satisfait le WIP (Voir Dedecker et al. [START_REF] Dedecker | On the weak invariance principle for nonadapted sequences under projective criteria[END_REF]2007]). Mentionnons également le critère projectif dû à Maxwell-Woodroofe. Pour ce faire, commençons par rappeler le résultat dû à Merlevède et al. [START_REF] Merlevède | Almost sure invariance principles via martingale approximation[END_REF]2012] :

∥S n (X) -S n (d)∥ 2 ≤ n 1/2 k≥n ∥E (S k |F 0 )∥ 2 n 3/2 .
On a donc :

Théorème (B6). Soit (X k ) k≥0 une suite strictement stationnaire et ergodique dans

L 2 (P), et (F k ) k≥0 la ltration engendré par (X k ) k≥0 . Si on a k≥1 ∥E (S k |F 0 )∥ 2 n 3/2 < ∞ , (14)
alors n -1/2 S n (X) converge en loi vers une loi normale centrée de variance

σ 2 = ∥X 0 ∥ 2 2 .
Tous les résultats susmentionnés sont sous la normalisation √ n. Mais il existe des processus où la normalisation √ n n'est pas susante pour obtenir le TCL ou le WIP. Par exemple, les processus linéaires. Considérons f une fonction C 1 lipschitzienne, et (ξ i ) i∈Z une suite de v.a. i.i.d. dans L 2 . Dénissons le processus (X k ) k∈Z avec

X k = f i≥0 ξ k-i i + 1 -E f i≥0 ξ k-i i + 1 .
Dans [START_REF] Dedecker | On the weak invariance principle for nonadapted sequences under projective criteria[END_REF]2007]

C 0 (s n ): lim n→∞ S n (X) s n - 1 √ n n i=1 d • T i 2 = 0 . C 1 (s n ):                      (a) ∥E [S n (X)|F 0 ]∥ 2 = o(s n ) , (b) ∥S n (X) -E [S n (X)|F n ]∥ 2 = o(s n ) , (c) lim n 1 ∧n 2 →∞ 1 n n i=1 √ n s n n-i k=1-i P 0 (X k ) -m 2 2 = 0 .
Si une des conditions est satisfaite et que la transformation T est ergodique, alors s -1 n S n (X) converge en loi vers σN (0, 1), où N (0, 1) est la loi normale centré réduite, et

σ 2 = E(d 2 ).
Ensuite, en utilisant cette approximation martingale, Dedecker et al. [START_REF] Dedecker | On the weak invariance principle for nonadapted sequences under projective criteria[END_REF]2007] ont montré le WIP pour les suites strictement stationnaires non nécessairement adaptées à la ltration (F i ) i∈Z .

Théorème (B8). Supposons que

C 1 (s n )(c) est satisfaite, et que pour tout t ∈ [0, 1] s [nt]
/s n est bornée. Supposons T ergodique et que la condition suivante est satisfaite :

C 2 (s n ):                                (a) ∥max 1≤k≤n |E [S k (X)|F 0 ]|∥ 2 = o(s n ) , (b) ∥max 1≤k≤n |S n (X) -E [S n (X)|F n ]|∥ 2 = o(s n ) , (c) pour certaine suite positive (u i ) i∈Z tell que √ n s n n i=-n u i est bornée, lim λ→∞ lim sup n→∞ √ n s n n i=-n E P 2 0 (X i ) u i 1 P 2 0 (X i )>Au 2 i = 0 , alors s -1 n S [nt] , t ∈ [0, 1] converge en loi dans D([0, 1]) vers {σB t } t∈[0,1] , avec σ = E (d 2 ).
Dans le chapitre 1, nous étendons ces deux résultats aux champs aléatoires strictement stationnaires non nécessairement adaptés à une ltration sous-jacente complètement commutante. Notons que le TCL pour les ortho-martingales a été établi par Volný [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF]2015]. Rappelons son énoncé dans le cadre des champs aléatoire indexés Il faut aussi noter que la convergence a toujours lieu même si aucune des transformation n'est ergodique. Cependant la somme partielle convergera vers un mélange des lois normales (voir [START_REF] Volný | Martingale-coboundary representation for stationary random elds[END_REF]2018]).

par Z 2 . Théorème (B9) (D.Volný 2015). Soit X ∈ L 2 (P) tel que (U k V ℓ X)
Dans le chapitre 3, nous nous intéresserons à établir des TCL et leurs version fonctionnelles pour des champs aléatoires à valeurs dans des espaces de Banach.

Mentionnons quelques résultats antérieurs sur ce sujet : Tout d'abord dans L 1 (S, µ) l'espace des fonctions réelles µ-intégrables avec µ σ-nie (on supposera L 1 (S, µ) séparable), Dédé [START_REF] Dede | An empirical central limit theorem in L1 for stationary sequences[END_REF]2009] a montré que pour une suite (d i ) i∈Z strictement stationnaire et ergodique de diérences de martingales à valeurs dans L 1 (S, µ) telle que,

S ∥d 0 (t)∥ 2 µ(dt) < ∞ , (15)
n -1/2 n k=1 d k satisfait le TCL dans L 1 (S, µ). Puis en utilisant une approximation martingale, elle a montré que si (X k ) k∈Z était une suite strictement stationnaire, ergodique à valeurs dans L 1 (S, µ) telle que

S ∥X 0 (t)∥ 2 µ(dt) < ∞ et i∈Z S ∥P i (X 0 (t))∥ 2 µ(dt) < ∞ , (16)
alors n -1/2 n k=1 X k convergeait en loi dans L 1 (S, µ) vers une variable gaussienne. Notons que [START_REF] Meyer | On the alternating double series[END_REF] est une condition de type Hannan. Plus récemment, Cuny [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF]2017] a établi des TCL et leurs version fonctionnelles pour des suites de v.a. à valeurs dans des espaces de Banach sous des conditions projectives. Il a d'abord montré le WIP pour les suites de diérences de martingale ergodiques à valeurs dans un espace de Banach séparable 2-lisse ou de cotype 2 (les espace L 1 sont de cotype 2). Puis, il a obtenu le WIP pour les suites strictement stationnaires et ergodiques à valeurs dans un espace de Banach de cotype 2 sous une condition de type Maxwell-Woodroofe généralisée. Dans L 1 (S, µ), cette condition s'écrit :

k≥1 S ∥E (S k (t)|F 0 )∥ 2 n 3/2 µ(dt) < ∞ .
Enn, Dedecker et Merlevède [START_REF] Dedecker | Moment bounds for dependent sequences in smooth Banach spaces[END_REF]2015] ont obtenu le WIP dans l'espace L 1 (S, µ) sous un critère projectif de type L 1 . Dans le chapitre 3, nous nous intéresserons à l'extension des résultats de Dédé [START_REF] Dede | An empirical central limit theorem in L1 for stationary sequences[END_REF] dans le cadre des champs aléatoires indexées par Z d et à l'obtention de la version fonctionnelle. On commencera par étendre aux ortho-martingales à valeurs dans des espaces de Banach 2-lisse ou de cotype 2, le WIP obtenu par Cuny [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF]

Chapter 2

On the weak invariance principle for non-adapted stationary random elds under projective criteria

In this chapter, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random elds non necessarily adapted, under dierent normalizations. To do so, we rst state sucient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP holds.

These projective criteria are in the spirit of the Hannan's condition and are well adapted to linear random elds with ortho-martingale innovations and which exhibit long memory.

Introduction

Let (X i ) i∈Z d , d ≥ 1, be a stationary random eld with zero mean and nite variance.

Let also S n be its associated partial sum with n = (n

1 • • • n d ) ∈ N d , dened as S n = n 1 i 1 =1 • • • n d i d =1 X i .
In this paper, we are interested in the central limit theorem and the weak invariance principle. In particular the WIP addresses the question of nding a positive sequence

s n such that s n → ∞ and S [n•t] s n t∈[0,1] d ⇒ {σW d (t)} t∈[0,1] d , (2.1.1) in D [0, 1] d equipped with the uniform topology. Here [n • t] means ([n 1 t 1 ] , • • • , [n d t d ])
and the notation " ⇒ " means the convergence in distribution. When the X k are functions of an i.i.d random eld, under the so-called Hannan's condition [START_REF] Hannan | Central limit theorems for time series regression, Z. Wahrscheinlichkeitstheor und verw[END_REF] generalized to Z d , Wang and Volný [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF] have established that (2.1.1) holds with

s 2 n = n 1 . . . n d . When d = 2
, their condition reads as

(i,j)∈Z 2 ∥P 0,0 (X i,j )∥ 2 < ∞, (2.1.2)
where P 0,0 is the projection operator dened by (2.1.4). We also refer to [START_REF] Zhang | On the quenched CLT for stationary random elds under projective criteria[END_REF] for a quenched version of the CLT under (2.1.2). The result stated in [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF] does not allow to consider models for which the normalization in the WIP does not have a linear growth with respect to n. Our paper is a step in this direction and aims at relaxing the condition (2.1.2) to still get the WIP. Our results can be viewed as the random eld counterparts of those established in Dedecker et al. [START_REF] Dedecker | On the weak invariance principle for nonadapted sequences under projective criteria[END_REF]. To do so, we start with a new ortho-martingale approximation using the notion of commuting ltration (note that there is no natural ordering of future and past in higher dimension).

Before giving a avor of the results obtained in this paper by considering a simple example of linear random eld with long memory, let us rst mention additional earlier results to [START_REF] Dedecker | On the weak invariance principle for nonadapted sequences under projective criteria[END_REF], involving projective type criteria and using the notion of orthomartingale. First, in [START_REF] Wang | A new condition for the invariance principle for stationary random elds[END_REF], Wang and Woodroofe proved the WIP for stationary random elds indexed by rectangular sets with the help of an m-dependent random eld approximation. Their condition is in term of conditional expectation and is in the spirit of the Maxwell-Woodroofe's condition in dimension 1 (see [START_REF] Maxwell | Central limit theorems for additive functionals of Markov chains[END_REF]). Then, as an application, they proved the WIP for ortho-martingales when they are functions of an i.i.d random eld. In [START_REF] Machkouri | A central limit theorem for stationary random elds[END_REF], using the same method (m-dependent random eld approximation) and the physical dependence measure, El Machkouri et al. proved a more general result in the sense that their result apply to a wider class of stationary random elds.

Concerning stationary ortho-martingales, a CLT has been obtained by Volný [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF]. More precisely, let us consider the case d = 2 and assume that (T i,j ) (i,j)∈Z 2 is a group of commuting probability preserving transformation of (Ω, A, P). Suppose in addition that h is a random variable in L 2 and that (h

• T i,j ) (i,j)∈Z 2 is an ortho- martingale (see denition 4.2.
2) with respect to a commuting ltration (F i,j ) (i,j)∈Z 2 (see denition 2.1.1). In this situation, it has been proved in [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF] that, as n 1 ∧n 2 → ∞,

(n 1 n 2 ) -1/2 n 1 i=1 n 2
j=1 h•T i,j converges in distribution to a centered gaussian random variable with variance ∥h∥ 2 2 provided that one of the transformation T 0,1 or T 1,0 is ergodic (here and along the paper the symbol a∧b stands for the minimum between a and b). Our strategy of proof is then to derive a new ortho-martingale approximation for S n /s n which combined with the CLT in [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF] will lead us to new projective criteria ensuring the convergence in law of S n /s n . To derive the corresponding WIP, we then state a new criterion ensuring the uniform integrability of s -2 n max k≤n S 2 k . To illustrate our results, let us consider the following linear elds in dimension 2 with long memory. Let X k,ℓ = i,j∈N 2 1 (i + 1)(j + 1) ξ 0,0 • T k-i,ℓ-j , with T the shift transformation dened as in subsection 1.1 and ξ 0,0 a centered random variable in

L 2 and such that E (ξ 0,0 |F -1,0 ) = E (ξ 0,0 |F 0,-1 ) = 0 a.s. ,
where in case of random elds with dimension 2. In Section 4, we apply our results to linear elds with ortho-martingale innovations. Then, in Section 5, we extend our results to higher dimension. Section 6 is devoted to the proofs of the main results. We end this section by giving some notations and denitions used all along the paper.

F i,j = σ {ξ h,k , h ≤ i, k ≤ j}.
Notations and Denitions. Let (Ω, A, P) be a probability space. To dene properly a stationary ltration, as in [START_REF] Zhang | On the quenched CLT for stationary random elds under projective criteria[END_REF], it is convenient to start with an auxiliary stationary process (ξ i,j ) i,j∈Z 2 and then to set

F i,j = σ{ξ h,k , h ≤ i, k ≤ j}. (2.1.3)
For all i, j, we also dene F ∞,∞ = (i,j)∈Z 2 F i,j , F i,-∞ = j∈Z F i,j , F -∞,j = i∈Z F i,j , and F -∞,-∞ = (i,j)∈Z 2 F i,j .

Dénition 2.1.1. The ltration (F i,j ) (i,j)∈Z 2 dened as above is said to be commuting

if ∀i, j, h, k ∈ Z, E [E [X|F i,j ] |F h,k ] = E [E [X|F h,k ] |F i,j ] = E [X|F i∧h,j∧k ] a.s. ,
provided all the above conditional expectations are well dened.

Note that a ltration dened by an independent and identically distributed (i.i.d) random eld is commuting. This kind of ltrations can also be constructed using stationary random elds with independent rows or columns (see [START_REF] Machkouri | A central limit theorem for stationary random elds[END_REF]). From now on, we assume that the ltration dened in (2.1.3) is commuting.

Next we introduce the projection operators dened by: for all i, j ∈ Z and

X ∈ L 1 (Ω), P i, j (X) = E [X|F i,j ] -E [X|F i,j-1 ] P i,j (X) = E [X|F i,j ] -E [X|F i-1,j ] .
Note that, if the ltration is commuting then P i,j (X) = P i,j • P i, j (X) = P i, j • P i,j (X).

Therefore P i,j (X) = E [X|F i,j ] -E [X|F i,j-1 ] -E [X|F i-1,j ] + E [X|F i-1,j-1 ] . (2.1.4)
Let us now introduce the shift operators as follows: on R Z 2 , let

T (1) ((x k,ℓ ) (k,ℓ)∈Z 2 ) = (x k+1,ℓ ) (k,ℓ)∈Z 2 , T (2) ((x k,ℓ ) (k,ℓ)∈Z 2 ) = (x k,ℓ+1 ) (k,ℓ)∈Z 2 .
(2.1.5)

We will denote

T i (1) • T j (2) (x k,ℓ ) (k,ℓ)∈Z 2 by T i,j ((x k,ℓ ) (k,ℓ)∈Z 2
). Note that the ltration dened above can also be rewritten F i,j = T -i,-j (F 0,0 ). Now we introduce a stationary random eld in the following way. For a real-valued measurable function f on R Z 2 , we dene

X 0,0 = f (ξ i,j ) (i,j)∈Z 2 and X k,ℓ = f T k,ℓ (ξ i,j ) (i,j)∈Z 2 = X 0,0 • T k,ℓ .
(2.1.6) Dénition 2.1.2. The stationary random eld (X i,j ) (i,j)∈Z 2 will be said to be regular if X 0,0 is F ∞,∞ -measurable, and

E (X 0,0 |F -∞,∞ ) = E (X 0,0 |F ∞,-∞ ) = E (X 0,0 |F -∞,-∞ ) = 0 a.s.
We will sometimes use the notation n 1 ,n 2 i,j=1 to mean the double sum n 1 i=1 n 2 j=1 . 20

Ortho-Martingale Approximation

Let (X i,j ) (i,j)∈Z 2 be a stationary random eld dened by (2.1.6) and let

S n 1 ,n 2 = n 1 i=1 n 2 j=1 X i,j and σ 2 n 1 ,n 2 = ∥S n 1 ,n 2 ∥ 2 2 .
Dénition 2.2.1. Let m be an integrable F 0,0 -measurable function. We say that (m • T i,j ) (i,j)∈Z 2 is a eld of ortho-martingale dierences with respect to a commuting ltration (F i,j ) (i,j)∈Z 2 , if for all i, j, k, ℓ ∈ Z, such that either k < i, or ℓ < j then

E (m • T i,j |F k,ℓ ) = 0 a.s. . In addition, M n 1 ,n 2 := n 1 i=1 n 2
j=1 m • T i,j is said to be an ortho-martingale.

Let (s n 1 ,n 2 ) n 1 ,n 2 ≥1 be a double indexed sequence of positive numbers such that

s n 1 ,n 2 → ∞ as n 1 ∧ n 2 → ∞.
Below we give an ortho-martingale approximation for the normalized partial sum

S n 1 ,n 2 /s n 1 ,n 2 .
Théorème 2.2.2. Let m be a square integrable F 0,0 -measurable function such that

E (m|F -1,0 ) = E (m|F 0,-1 ) = 0 a.s. and let R n 1 ,n 2 = ∥E [S n 1 ,n 2 |F n 1 ,0 ]∥ 2 +∥E [S n 1 ,n 2 |F 0,n 2 ]∥ 2 + ∥E [S n 1 ,n 2 |F 0,0 ]∥ 2 .
Then the following conditions are equivalent

C 0 (s n 1 ,n 2 ): lim n 1 ∧n 2 →∞ S n 1 ,n 2 s n 1 ,n 2 - 1 √ n 1 n 2 n 1 i=1 n 2 j=1 m • T i,j 2 = 0 . C 1 (s n 1 ,n 2 ):                      (a) R n 1 ,n 2 = o(s n 1 ,n 2 ), (b) ∥S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]∥ 2 = o(s n 1 ,n 2 ), (c) lim n 1 ∧n 2 →∞ 1 n 1 n 2 n 1 ,n 2 i,j=1 √ n 1 n 2 s n 1 ,n 2 n 1 ,n 2 h,k=1 P 0,0 (X h-i,k-j ) -m 2 2 = 0 .
If one of the conditions is satised and T (1) or T (2) is ergodic, then s -1 n 1 ,n 2 S n 1 ,n 2 converges in distribution to σN , where N is a standard Gaussian random variable, and

σ 2 = E(m 2 ). Remark 2.2.3. Note that if E [m 2 ] < ∞ then the convergence in law of M n 1 ,n 2 √ n 1 n 2
always takes place, but to a mixture of normal laws (see [START_REF] Volný | On limit theorems for elds of martingale dierences[END_REF]). However, the additional condition that at least one of the transformation is ergodic ( T (1) or T (2) ) guarantees the convergence towards a normal law. For more detailed results see [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF].

Proposition 2.2.4. If C 1 (σ n 1 ,n 2 )(a) and (b) hold then σ n 1 ,n 2 √ n 1 n 2
is a two-parameter slowly varying function (2p-svf) in the following sense : for any non negative integers k and ℓ

lim n 1 ∧n 2 →∞ σ 2 kn 1 ,ℓn 2 σ 2 n 1 ,n 2 = k • ℓ.
(2.2.1)

In addition, if σ n 1 ,n 2 → ∞ as n 1 ∧ n 2 goes to innity and if

lim n 1 ∧n 2 →∞ σ 1,n 2 σ n 1 ,n 2 = 0, lim n 1 ∧n 2 →∞ σ n 1 ,1 σ n 1 ,n 2 = 0, (2.2.2)
then,

lim x∧y→∞ σ 2 [kx],[ℓy] σ 2 [x],[y] = k • ℓ. (2.2.3)
Next we give a similar remark carried from its one dimensional version (see Remark 3 in [START_REF] Dedecker | On the weak invariance principle for nonadapted sequences under projective criteria[END_REF]).

Remark 2.2.5.

If C 0 (s n 1 ,n 2 ) holds then s -2 n 1 ,n 2 σ 2 n 1 ,n 2 converges to E (m 2 ). Con- sequently, if E (m 2 ) > 0 then C 0 (σ n 1 ,n 2 ) holds with m ′ = m/ ∥m∥ 2 . Therefore C 1 (σ n 1 ,n 2 ) (a)
∧ n 2 → ∞, √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=-n 2 P 0,0 (X h,k ) -→ m in L 2 ,
(2.2.4)

n 1 n 2 j=1 n 1 h=-n 1 n 2 k=j P 0,0 (X h,k ) 2 2 + n 1 h=-n 1 -j k=-n 2 P 0,0 (X h,k ) 2 2 = o(s 2 n 1 ,n 2 ), (2.2.5
)

n 1 i=1 n 2 j=1 n 1 h=i n 2 -j k=1-j P 0,0 (X h,k ) 2 2 + -i h=-n 1 n 2 -j k=1-j P 0,0 (X h,k ) 2 = o(s 2 n 1 ,n 2 ). (2.2.6)
In particular if X 0,0 is F 0,0 -measurable and

s n 1 ,n 2 √ n 1 n 2 = h 1 (n 1 )h 2 (n 2 )
, with h 1 and h 2 two one-parameter slowly varying functions, then C 0 (s n 1 ,n 2 ) is satised as soon as:

R n 1 ,n 2 = o(s n 1 ,n 2 ) and

√ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) → m in L 2 .
(2.2.7)

As a consequence of the previous proposition, we obtain the following corollary:

Corollary 2.2.7. Consider the following conditions:

n 1 i=-n 1 n 2
j=-n 2 P 0,0 (X i,j ) -→ m in L 2 , and

∥S n 1 ,n 2 ∥ 2 √ n 1 n 2 -→ ∥m∥ 2 , as n 1 ∧ n 2 → ∞, (2.2.8)
X 0,0 is regular and

(i,j)∈Z 2 ∥P 0,0 (X i,j )∥ 2 < ∞.
(2.2.9)

We have the implications (2.2.9)

⇒ (2.2.8) ⇒ C 1 ( √ n 1 n 2 ).
Note that in dimension one, (2.2.9) is the so-called Hannan's condition. The condition (2.2.8) can be viewed as a non-adapted version of Theorem 5 of [START_REF] Peligrad | Martingale approximations for random elds[END_REF].

Weak invariance principle

We start this section by giving sucient conditions for the sequence s -2

n 1 ,n 2 max 1≤k≤n 1 1≤l≤n 2 S 2 k,l (n 1 ,n 2 )∈N 2
to be uniformly integrable. With this aim, let us introduce the following notation:

R n 1 ,n 2 (S k,l ) = E(S k,l |F n 1 ,0 ) + E(S k,l |F 0,n 2 ) -E(S k,l |F 0,0 ) . Proposition 2.3.1. The sequence s -2 n 1 ,n 2 max 1≤k≤n 1 1≤l≤n 2 S 2 k,l (n 1 ,n 2 )∈N 2
is uniformly integrable if the following conditions are satised:

C 2 (s n 1 ,n 2 ):                                      (a) sup 1≤k≤n 1 1≤l≤n 2 |R n 1 ,n 2 (S k,l )| 2 = o(s n 1 ,n 2 ), (b) sup 1≤k≤n 1 1≤l≤n 2 |S k,l -E [S k,l |F n 1 ,n 2 ]| 2 = o(s n 1 ,n 2 ),
(c) for some positive sequence (u i,j ) i,j∈Z 2 such that

√ n 1 n 2 s n 1 ,n 2 n 1 i=-n 1 n 2 j=-n 2 u i,j is bounded, lim λ→∞ lim sup n 1 ∧n 2 →∞ √ n 1 n 2 s n 1 ,n 2 n 1 i=-n 1 n 2 j=-n 2 E U 2 0,0 (i, j, λ) = 0,
where U 2 0,0 (i, j, λ) := P 2 0,0 (X i,j ) u i,j 1 P 2 0,0 (X i,j )>λu 2 i,j . Remark 2.3.2. Note that if the rst part of C 2 (s n 1 ,n 2 )(c) holds then its second part does as soon as P 0,0 (X i,j ) u i,j

(i,j)∈Z 2
is a square uniformly integrable family.

We give now sucient conditions for C 2 (s n 1 ,n 2 )(a) and (b) to hold.

Proposition 2.3.3. C 2 (s n 1 ,n 2 )(a) and (b) hold as soon as X 0,0 is regular and

C 3 (s n 1 ,n 2 ):                    (a) n 1 u=1 n 2 v=1 |i|≥u |j|≥v ∥P 0,0 (X i,j )∥ 2 2 = o(s n 1 ,n 2 ), (b) √ n 1 n 1 -1 i=1-n 1 n 2 v=1 |j|≥v ∥P 0,0 (X i,j )∥ 2 2 = o(s n 1 ,n 2 ), (c) √ n 2 n 1 u=1 n 2 -1 j=1-n 2 |i|≥u ∥P 0,0 (X i,j )∥ 2 2 = o(s n 1 ,n 2 ).
In addition, in the adapted case, C 3 (s n 1 ,n 2 ) holds provided

C 3 (s n 1 ,n 2 ) :            (a) √ n 1 n 1 u=1 n 2 v=1 1 √ u ∥E (X u,v |F 0,0 )∥ 2 = o(s n 1 ,n 2 ), (b) √ n 2 n 1 u=1 n 2 v=1 1 √ v ∥E (X u,v |F 0,0 )∥ 2 = o(s n 1 ,n 2 ).
Remark 2.3.4. If s 2 n 1 ,n 2 is regular enough, so for instance if there exist α and β in ]0, 1[ such that

n 1-α 1 n 1-β 2 (n 1 + n 2 ) s 2 n 1 ,n 2 → ∞ as n 1 ∧ n 2 → ∞,
then Hölder's inequality combined with Kronecker's Lemma for double indexed sequences implies that

C 3 (s n 1 ,n 2 ) is satised as soon as u,v≥1 uv s 2 u,v ∥E (X u,v |F 0,0 )∥ 2 2 < ∞.
According to the lemma stated in page 88 in [START_REF] Billingsley | Convergence of probability measures[END_REF], the uniform integrability of

s -2 n 1 ,n 2 max 1≤k≤n 1 1≤l≤n 2 S 2 k,l (n 1 ,n 2 )∈N 2
implies the tightness of {s -1

n 1 ,n 2 S [n 1 t 1 ],[n 2 t 2 ] , (t 1 , t 2 ) ∈ [0, 1] 2 } in D([0, 1] 2
t ∈ [0, 1] 2 . If C 1 (s n 1 ,n 2 )(c) and C 2 (s n 1 ,n 2 ) hold and T (1) or T (2) is ergodic, then s -1 n 1 ,n 2 S [n 1 t 1 ],[n 2 t 2 ] , t ∈ [0, 1] 2 converges in distribution in D([0, 1] 2 ) to σW 2 where W 2 is a 2-parameter Brownian sheet and σ = E (m 2 ).
Remark 2.3.6. The proof reveals that if we assume that C 1 (s n 1 ,n 2 )(c) holds with m such that E (m 2 ) > 0, then we do not need to assume the boundedness of (s

[n 1 t 1 ],[n 2 t 2 ] /s n 1 ,n 2 ) t∈[0,1] 2
in the statement of the previous theorem.

Application to linear elds of ortho-martingales

Dene X 0,0 = i,j∈Z 2 a i,j ξ 0,0 • T -i,-j where (a i,j ) (i,j)∈Z 2 a double-indexed sequence of real numbers in ℓ 2 , and ξ 0,0 is a regular F 0,0 -measurable function in L 2 such that

E (ξ 0,0 |F -1,0 ) = E (ξ 0,0 |F 0,-1 ) = 0 a.s. . Let ξ k,ℓ = ξ 0,0 • T k,ℓ and S n 1 ,n 2 = n 1 k=1 n 2 ℓ=1 X k,ℓ = n 1 k=1 n 2 ℓ=1 (i,j)∈Z 2 a i,j ξ k-i,ℓ-j .
Corollary 2.4.1. Let ξ 0,0 , (a k,ℓ ) k,ℓ∈Z 2 and X 0,0 be dened as above. We then dene the double-indexed sequence

s n 1 ,n 2 = √ n 1 n 2 n 1 i=-n 1 n 2 j=-n 2 a i,j
. Assume the following conditions:

lim sup n 1 ∧n 2 →∞ n 1 k=-n 1 n 2 ℓ=-n 2 |a k,ℓ | n 1 k=-n 1 n 2 ℓ=-n 2 a k,ℓ < ∞ , (2.4.1)
n 1 u=1 n 2 v=1 |i|≥u |j|≥v a 2 i,j = o(s n 1 ,n 2 ) , (2.4.2)
√ n 1 n 1 -1 i=1-n 1 n 2 v=1 |j|≥v a 2 i,j = o(s n 1 ,n 2 ) , (2.4.3)
and

√ n 2 n 1 u=1 n 2 -1 j=1-n 2 |i|≥u a 2 i,j = o(s n 1 ,n 2 ) . (2.4.4) Then s -1 n 1 ,n 2 S [n 1 t 1 ][n 2 t 2 ] , (t 1 , t 2 ) ∈ [0, 1] 2 converges in distribution in D([0, 1] 2 ) to E ξ 2 0,0 W 2 , where W 2 is a 2-parameter Brownian sheet. Remark 2.4.2. If Condition (2.4.1) is satised and i,j∈Z |a i,j | < ∞ instead of assuming (2.4.2)-(2.4.4
), then the conclusion of Corollary 2.4.1 follows from Theorem 5.1 in [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF] since, according to Condition (2.4.1), as

n 1 , n 2 → ∞, s n 1 ,n 2 / √ n 1 n 2 → i,j∈Z a i,j > 0.
Comment 2.4.3. Condition (2.4.1) of the above corollary does not allow the following possibility:

n 1 k=-n 1 n 2 ℓ=-n 2 |a k,ℓ | diverges but n 1 k=-n 1 n 2
ℓ=-n 2 a k,ℓ converges. For instance let us consider the following sequence: a i,j = (-1) i+j (i + j) -2 for i, j ≥ 1 and 0 otherwise, then Corollary 2.4.1 does not apply. More generally, this is also the case for double alternating series i,j a i,j (in the sense that each row and column is an alternating simple series) that are in addition monotonic which means that |a i,j | ≤ |a n,m | for i ≥ n and j ≥ m. Indeed such series are not absolutely convergent if i,j |a i,j | = ∞ but convergent as soon as |a i,j + a i+1,j | ≥ |a i,j+1 + a i+1,j+1 | for all i, j (see Meyer [START_REF] Meyer | On the alternating double series[END_REF] for more details) and then, in this situation, condition (2.4.1) fails. However, by simple algebra, we infer that, for such series, condition (2.2.7) holds with s n 1 ,n 2 ∼ √ n 1 n 2 i,j≥0 a i,j (and then s -1 n,n S n,n converges in distribution to a Gaussian random variable) as soon as

u,v≥0 n 1 k=1 n 2 ℓ=1 a k+u,ℓ+v 2 + n 1 u=1 n 1 k=1 n 2 ℓ=1 a k+u,ℓ+v 2 = o(n 1 n 2 )
and

n 1 u=1 v≥0 n 1 k=u n 2 ℓ=1 a k+u,ℓ+v 2 + n 2 v=1 u≥0 n 1 k=1 n 2 ℓ=v a k+u,ℓ+v 2 = o(n 1 n 2 ).
Clearly, both of the above conditions hold if u,v≥0 k≥u ℓ≥v a k,ℓ 2 < ∞ which is satised as soon as i,j≥0 a 2 i,j < ∞ (to see this, use an Abel transformation and the monotonicity conditions on the sequence (a i,j )).

Examples

Example 2.4.4. As quoted in the introduction, the double indexed sequence

(a k,ℓ ) (k,ℓ)∈Z 2 dened by a k,ℓ = 1 k + 1 1 ℓ + 1
for k ≥ 0 and ℓ ≥ 0, and 0 otherwise, satises the condition of Corollary 2.4.1 with

s n 1 ,n 2 ∼ √ n 1 n 2 log(n 1 ) log(n 2 ).
Example 2.4.5. Let us consider now another example in the same spirit but for which the normalizing sequence s n 1 ,n 2 is not a product of a function of n 1 times a function

of n 2 . The double indexed sequence (a k,ℓ ) k,ℓ∈Z 2 is this time dened by a k,ℓ = 1 (k + ℓ) 2
for k and ℓ > 0, and 0 otherwise. In this case the conditions of Corollary 2.4.1 are satised with

s n 1 ,n 2 ∼ √ n 1 n 2 log n 1 n 2 n 1 + n 2 .
Comment 2.4.6. It is also possible to apply our Theorem 2.3.5 to non linear random elds. For instance, let us consider the following Volterra process

X k,ℓ = i,j≥0 ξ k-i,ℓ-j (α i,j ξ k-i-1,ℓ-j + β i,j ξ k-i,ℓ-j-1 ) ,
where (ξ k,ℓ ) k,ℓ is a sequence of ortho-martingale dierences. In this case, setting

s n 1 ,n 2 = √ n 1 n 2 n 1 i=0 n 2 j=0 (α i,j + β i,j
) and assuming that there exist reals α and β such that

lim n 1 ∧n 2 →∞ n 1 i=0 n 2 j=0 α i,j n 1 i=0 n 2 j=0 (α i,j + β i,j ) = α and lim n 1 ∧n 2 →∞ n 1 i=0 n 2 j=0 β i,j n 1 i=0 n 2 j=0 (α i,j + β i,j ) = β , (2.4.5)
one sees that condition (2.2.4) holds with m = ξ 0,0 αξ -1,0 + βξ 0,-1 . Moreover, following the lines of the proof of Corollary 2.4.1, we infer that, if in addition to (2.4.5), we assume that

lim n 1 ∧n 2 →∞ n 1 i=0 n 2 j=0 (|α i,j | + |β i,j |) n 1 i=0 n 2 j=0 (α i,j + β i,j ) < ∞ and conditions (2.4.2)-(2.4.4) are satised with α 2 i,j + β 2 i,j replacing a 2 i,j in the numer- ators, then s -1 n 1 ,n 2 S [n 1 t 1 ][n 2 t 2 ] , (t 1 , t 2 ) ∈ [0, 1] 2 converges in distribution in D([0, 1] 2 ) to c α,β W 2 , where W 2 is a 2-parameter Brownian sheet and c 2 α,β = E(m 2 ).

Extension to higher dimension d > 2

In this section, we extend our results to dimension d > 2. To do so, we rst introduce some notations in dimension d. Let i, j be two elements of Z d . For i := (i 1 , . . . , i d ), and j := (j 1 , . . . , j d ), we set

i + j := (i 1 + j 1 , . . . , i d + j d ) i ≤ j mean that i k ≤ j k for all 1 ≤ k ≤ d i ∧ j := (i 1 ∧ j 1 , . . . , i d ∧ j d ) |n| := (n 1 × • • • × n d )
Note also that the integers of Z d will be in bold, for example 1 = (1, . . . , 1) ∈ Z d .

Next for n = (n 1 , . . . , n d ), and for every 1 ≤ k ≤ d, we set n k = (0, . . . , n k , . . . , 0),

n k 1 k 2 = (0, . . . , 0, n k 1 , 0, . . . , 0, n k 2 , 0, . . . , 0) and so on. Also n -k = (n 1 , . . . , n k-1 , 0, n k+1 , . . . , n d ).
As in dimension 2, let F i = σ(ξ j , j ≤ i) where (ξ i ) i∈Z d is an auxiliary stationary process. We assume that the ltration is commuting. The shift transformation

T (1) , • • • , T (d) is dened as follows: on R Z d , for 1 ≤ j ≤ d, T (j) ((x k ) k∈Z d ) = (x k 1 ,••• ,k j-1 ,k j +1,k j+1 ,••• ,k d ) k∈Z d .
(2.5.1)

For i = (i 1 , • • • , i d ), we set T i 1 (1) • • • • • T i d (d) ((w k ) k∈Z d ) by T i ((w k ) k∈Z d .
Next, for a real valued measurable function f on R Z d , we dene

X 0 = f (ξ i ) i∈Z d , and X k = f T k (ξ i ) i∈Z d = X 0 • T k .
Let S n be its associated partial sum dened as

S n = n 1 i 1 =1 • • • n d i d =1 X i .
We now clarify the rest R n (S n ) in the decomposition of S n with the help of the projective operators. We have

S n = n 1 k 1 =1 • • • n d k d =1 P k (S n ) + R n (S n ), where R n (S i ) = (-1) d-1 E [S i |F 0 ] + (-1) d-2 d k=1 E S i |F n k + (-1) d-3 d-1 k 1 =1 d k 2 >k 1 E S i |F n k 1 k 2 +(-1) d-4 d-2 k 1 =1 d-1 k 2 >k 1 d k 3 >k 2 E S i |F n k 1 k 2 k 3 + • • • +(-1) d-d d-(d-1)+1 k 1 =1 3 k 2 >k 1 • • • d-1 k d-2 >k d-3 d k d-1 >k d-2 E S i |F n k 1 k 2 ..k d-1 .
To simplify its expression, let us introduce some specic notations. We dene the set D of integers from 1 to d,

(D := (1, ..., d)). Next for any 1 ≤ k ≤ d, Q k will designate a set of the type {i 1 , . . . , i k } with i u ̸ = i v for u ̸ = v and i u ∈ D for any 1 ≤ u ≤ k.
Notice that for k xed, there exist dierent Q k since there are d k ways to chose a set of k dierent integers among D. Thus we will numerate each combination of

Q k by Q ℓ k for 1 ≤ ℓ ≤ d k
, and these combinations will be numerated in the natural increasing way. For example for k = 1,

Q 1 1 = {1} , Q 2 1 = {2} . . . Q d 1 = {d} and for k = 2, Q 1 2 = {1, 2} , Q 2 2 = {1, 3} . . . Q d-1 2 = {1, d}, Q (d-1)+1 2 = {2, 3} . . . Q (d-1)+(d-2)+1 2 = {3, 4}. In addition (Q ℓ k ) C will denote the complement of Q ℓ k in D. For Q ℓ k := {e 1 . . . e k }, we denote F Q ℓ k by F n e 1 ...e k . Hence, setting u k,d = d k , we have R n (S i ) = (-1) d-1 E [S i |F 0 ] + d-1 k=1 (-1) d-1-k u k,d ℓ=1 E S i |F Q ℓ k . (2.5.2) 2.5.1 Ortho-Martingale approximation and CLT in dimension d ≥ 2 Théorème 2.5.1. Let m be a F 0 -measurable function such that, E (m|F -1,0...0 ) = E (m|F 0,-1,0...0 ) = • • • = E (m|F 0...0,-1 ) = 0.
Then the two following conditions are equivalent

C d 0 (s n ): lim n→∞ S n s n - 1 √ n n 1 i 1 =1 • • • n d i d =1 m • T i 2 = 0 . C d 1 (s n ):                                (a) for every 0 ≤ k ≤ d -1, and 1 ≤ ℓ ≤ d k E S n |F Q ℓ k 2 = o(σ n ), (b) ∥S n -E [S n |F n ]∥ 2 = o(s n ), (c) lim n→∞ 1 n n 1 i 1 =1 • • • n d i d =1 √ n s n n 1 k 1 =1 • • • n d k d =1 P 0 (X k-i ) -m 2 2 = 0.
If one of the conditions is satised and one of

T (1) , • • • , T (d) is ergodic, then s -1 n S n
converges in distribution to σN , where N is a standard gaussian random variable, and

σ 2 = E(m 2 ). Remark 2.2.4 writes in dimension d > 2 as follows: Proposition 2.5.2. If C d 1 (s n )(a) and (b) hold, then lim n→∞ σ 2 k 1 n 1 ,...,k d n d σ 2 n 1 ,...,n d = lim n→∞ E S 2 k 1 n 1 ,...,k d n d E S 2 n 1 ,...,n d = k 1 • • • k d . (2.5.3) In addition, if σ n 1 ,...,n d → ∞ as n 1 ∧ . . . ∧ n d → ∞ and if for every 1 ≤ k ≤ d -1, and 1 ≤ ℓ ≤ u k,d , we have lim n 1 ∧...∧n d →∞ σ Q ℓ k σ n 1 ,...,n d = 0, (2.5.4)
where for example

σ Q 1 1 = σ 1,n 2 ,...,n d . Then, lim x 1 ∧...∧x d →∞ σ 2 [k 1 x 1 ],...,[k d x d ] σ 2 [x 1 ],...,[x d ] = k 1 • • • k d . (2.5.5)
As in dimension 2, we now give a sucient condition to prove

C d 1 (s n )(c). Proposition 2.5.3. Condition C d 1 (s n )(c) holds as soon as |n| s n n 1 k 1 =-n 1 • • • n d k d =-n d P 0 (X k ) -→ m in L 2 , (2.5.6)
and

d k=1 n 1 u 1 =1 • • • n d u d =1 nq iq=-nq q∈Q 1 k-1 n k i k =u k + -u k i k =-n k np-up ip=1-up p∈(Q 1 k ) C P 0 (X i ) 2 2 = o(s 2 n ), (2.5.7)
with

Q 1 k = {1, 2, 3, . . . , k} and Q 1 0 = (Q 1 d ) C = ∅.
In particular if X 0 is F 0 -measurable and

s n |n| = h 1 (n 1 ) . . . h d (n d ) with h k a one-parameter slowly varying function for k = 1, ..., d, then C d 0 (s n ) is satised as soon as: |n| s n n 1 k 1 =0 • • • n d k d =0 P 0 (X k ) -→ m in L 2 , (2.5.8)
and

C d 0 (s n )(a) is satised.
As a consequence, we get a generalized version of Corollary 2.2.7.

Corollary 2.5.4. Considering the following conditions

i∈Z d P 0 (X i ) -→ L 2 m,
and

∥S n ∥ 2 √ n -→ ∥m∥ 2 , (2.5.9)
X 0 is regular and

i∈Z d ∥P 0 (X i )∥ 2 < ∞. (2.5.10)
We have the implications (2.5.10) ⇒ (2.5.9) ⇒ C d

2.5.2

Weak invariance principle in dimension d ≥ 2 Proposition 2.5.5. The sequence s -2 n max 1≤i≤n S 2 i n∈N d is uniformly integrable if and only if the following conditions are satised :

C d 2 (s n ):                              (a) sup 1≤i≤n |R n (S i )| 2 = o(s n ), (b) sup 1≤i≤n |S i -E [S i |F n ]| 2 = o(s n ).
(c) for some positive sequence

(u i ) i∈Z d such that |n| s n n i=-n u i is bounded, lim λ→∞ lim sup n 1 ∧•••∧n d →∞ |n| s n n i=-n E P 2 0 (X n ) u i 1 P 2 0 (X i )>λu 2 i = 0.
Remark 2.3.3 can also be adapted to dimension d.

Remark 2.5.6.

Condition C d 2 (s n )(a)
and (b) hold as soon as, X 0 is regular and

n 1 u 1 =1 • • • n d u d =1 |i 1 |≥u 1 • • • |i d |≥u d ∥P 0 (X i )∥ 2 2 = o(s n ), (2.5.11)
and for every

1 ≤ k ≤ d -1, and 1 ≤ ℓ ≤ d k , q∈Q ℓ k √ n q nq-1 iq=1-nq q∈Q ℓ k np ip=1 p∈(Q ℓ k ) C +∞ ip=up p∈(Q ℓ k ) C ∥P 0 (X i )∥ 2 2 = o(s n ), (2.5.12
)

q∈Q ℓ k √ n q nq-1 iq=1-nq q∈Q ℓ k np ip=1 p∈(Q ℓ k ) C -up ip=-∞ p∈(Q ℓ k ) C ∥P 0 (X i )∥ 2 2 = o(s n ), (2.5.13)
Next, Theorem 2.5.1 together with Proposition 2.5.5 give the following weak invariance principle Théorème 2.5.7. Assume C d 2 (s n ) holds and that one of

T (1) , • • • , T (d) is ergodic. In addition suppose that s [n•t] /s n is bounded for any t ∈ [0, 1] d and E (m 2 ) < ∞. If C d 1 (s n )(c) holds, then s -1 n S [n•t] , t ∈ [0, 1] d converges in distribution in D([0, 1] d ) to σW d where W d is a d-parameter Brownian sheet and σ = E (m 2 ).
The proofs of the results stated in this section are of the same avor as those developed in Section 6 when d = 2 but with more tedious computations due to the higher dimension. They are therefore omitted and left to the reader. Note rst that the following decomposition holds :

S n 1 ,n 2 = S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] + R n 1 ,n 2 (S n 1 ,n 2 ) + E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 ), (2.6.1)
with

R n 1 ,n 2 (S n 1 ,n 2 ) = E(S n 1 ,n 2 |F n 1 ,0 ) + E(S n 1 ,n 2 |F 0,n 2 ) -E(S n 1 ,n 2 |F 0,0 ).
Then we denote n 1 i=1 n 2 j=1 m • T i,j by M n 1 ,n 2 . Note rst that with the denition of m, we have that

E (M n 1 ,n 2 |F 0,0 ) = E (M n 1 ,n 2 |F n 1 ,0 ) = E (M n 1 ,n 2 |F 0,n 2 ) = 0. Therefore recalling that R n 1 ,n 2 = ∥E [S n 1 ,n 2 |F n 1 ,0 ]∥ 2 + ∥E [S n 1 ,n 2 |F 0,n 2 ]∥ 2 + ∥E [S n 1 ,n 2 |F 0,0 ]∥ 2 , we derive that R n 1 ,n 2 ≤ 3 ∥S n 1 ,n 2 -M n 1 ,n 2 ∥ 2 .
Hence C 0 (s n 1 ,n 2 ) entails C 1 (s n 1 ,n 2 )(a). Next, using (2.6.1) and orthogonality, we have :

S n 1 ,n 2 s n 1 ,n 2 - M n 1 ,n 2 √ n 1 n 2 2 2 = S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] s n 1 ,n 2 2 2 + R n 1 ,n 2 (S n 1 ,n 2 ) s n 1 ,n 2 2 2 + E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 ) s n 1 ,n 2 - M n 1 ,n 2 √ n 1 n 2 2 2 := A 2 + B 2 + C 2 .
(2.6.2)

Therefore C 0 (s n 1 ,n 2 ) implies C 1 (s n 1 ,n 2 )(b)
and C → 0. Note now that the following decomposition is valid:

E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 ) = n 1 i=1 n 2 j=1 P i,j (S n 1 ,n 2 ). (2.6.3)
For more details concerning the decomposition we refer to [START_REF] Zhang | On the quenched CLT for stationary random elds under projective criteria[END_REF]. Using (2.6.3), it follows that

C = 1 √ n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 n 1 h=1 n 2 k=1 P i,j (X h,k ) s n 1 ,n 2 -m • T i,j 2 .
Hence, by orthogonality, we get

C 2 = 1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 n 1 h=1 n 2 k=1 P i,j (X h,k ) s n 1 ,n 2 -m • T i,j 2 2
.

Using stationarity, we then derive

C 2 = 1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 n 1 h=1 n 2 k=1 P 0,0 (X h-i,k-j ) s n 1 ,n 2 -m 2 2 . (2.6.4) Therefore, C → 0 is equivalent to C 1 (s n 1 ,n 2 )(c). This ends the proof of C 0 (s n 1 ,n 2 ) ⇒ C 1 (s n 1 ,n 2 ).
For the converse, we use (2.6.2), (2.6.4) and the fact that

∥R n 1 ,n 2 (S n 1 ,n 2 )∥ 2 ≤ R n 1 ,n 2 . Now if either C 0 (s n 1 ,n 2 ) or C 1 (s n 1 ,n 2 )
σ 2 kn 1 ,ℓn 2 = ∥S n 1 ,ℓn 2 ∥ 2 2 + k i=2 S in 1 ,ℓn 2 -S (i-1)n 1 ,ℓn 2 2 2 + 2B, (2.6.5)
where

B = E S n 1 ,ℓn 2 k i=2 S in 1 ,ℓn 2 -S (i-1)n 1 ,ℓn 2 .
Hence by stationarity,

σ 2 kn 1 ,ℓn 2 = k ∥S n 1 ,ℓn 2 ∥ 2 2 + 2 k-1 i=2 k j=i+1 E S n 1 ,ℓn 2 S (j-i+1)n 1 ,ℓn 2 -S (j-i)n 1 ,ℓn 2 + 2B := k ∥S n 1 ,ℓn 2 ∥ 2 2 + 2A + 2B.
We shall then focus on ∥S n 1 ,ℓn 2 ∥ 2 2 . Proceeding as before, we have

∥S n 1 ,ℓn 2 ∥ 2 2 = ℓ ∥S n 1 ,n 2 ∥ 2 2 + 2 ℓ-1 i=2 ℓ j=i+1 E S n 1 ,n 2 S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 + 2E S n 1 ,n 2 ℓ i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 := ℓ ∥S n 1 ,n 2 ∥ 2 2 + 2C + 2D.
Therefore,

σ 2 kn 1 ,ℓn 2 = k • ℓ ∥S n 1 ,n 2 ∥ 2 2 + 2A + 2B + 2C + 2D.
Hence to prove (2.2.1), it is sucient to show that A, B, C and D are o(σ 2 n 1 ,n 2 ). We rst handle D and write

|D| ≤ E (S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]) ℓ i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 + E E [S n 1 ,n 2 |F n 1 ,n 2 ] ℓ i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 := D 1 + D 2 .
By stationarity and using C 1 (σ n 1 ,n 2 )(b) we infer that

D 1 ≤ ∥S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]∥ 2 ℓ i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 2 ≤ (ℓ -1) ∥S n 1 ,n 2 ∥ 2 ∥S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]∥ 2 = o(σ 2 n 1 ,n 2 ).
On another hand, using stationarity and

C 1 (σ n 1 ,n 2 )(a) we get D 2 ≤ ∥S n 1 ,n 2 ∥ 2 E ℓ i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 |F n 1 ,n 2 2 ≤ (ℓ -1) ∥S n 1 ,n 2 ∥ 2 ∥E (S n 1 ,n 2 |F n 1 ,0 )∥ 2 = o(σ 2 n 1 ,n 2 ).
Thus D = o(σ 2 n 1 ,n 2 ). We handle now the quantity C and write

|C| ≤ ℓ-1 i=2 ℓ j=i+1 E (S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]) S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 + ℓ-1 i=2 ℓ j=i+1 E E [S n 1 ,n 2 |F n 1 ,n 2 ] S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 := C 1 + C 2 .
Using stationarity C 1 (σ n 1 ,n 2 )(b), we get

C 1 ≤ ℓ-1 i=2 ℓ j=i+1 ∥S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]∥ 2 S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 2 ≤ ℓ 2 ∥S n 1 ,n 2 ∥ 2 ∥S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]∥ 2 = o(σ 2 n 1 ,n 2 ).
Next, by stationarity and C 1 (σ n 1 ,n 2 )(a) we infer that,

C 2 ≤ ℓ-1 i=2 ℓ j=i+1 ∥S n 1 ,n 2 ∥ 2 E S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 |F n 1 ,n 2 ≤ ℓ 2 ∥S n 1 ,n 2 ∥ 2 ∥E (S n 1 ,n 2 |F n 1 ,0 )∥ 2 = o(σ 2 n 1 ,n 2 ). Recall that, ∥S n 1 ,ℓn 2 ∥ 2 2 = ℓ ∥S n 1 ,n 2 ∥ 2 2 + 2C + 2D. Thereby, lim n 1 ∧n 2 →∞ ∥S n 1 ,ℓn 2 ∥ 2 2 ∥S n 1 ,n 2 ∥ 2 2 = ℓ. (2.6.6)
We handle now the quantity B. We infer that

|B| ≤ E (S n 1 ,ℓn 2 -E [S n 1 ,ℓn 2 |F n 1 ,ℓn 2 ]) k i=2 S in 1 ,ℓn 2 -S (i-1)n 1 ,ℓn 2 + E E S n 1 ,ℓn 2 |F n1,ℓn 2 k i=2 S in 1 ,ℓn 2 -S (i-1)n 1 ,ℓn 2 := B 1 + B 2 .
Using stationarity and C 1 (σ n 1 ,n 2 )(b) and (2.6.6), we get

B 1 ≤ ∥S n 1 ,ℓn 2 -E [S in 1 ,ℓn 2 |F in 1 ,ℓn 2 ]∥ 2 k i=2 S in 1 ,ℓn 2 -S (i-1)n 1 ,ℓn 2 2 ≤ (k -1) ∥S n 1 ,ℓn 2 -E [S in 1 ,ℓn 2 |F in 1 ,ℓn 2 ]∥ 2 ∥S n 1 ,ℓn 2 ∥ 2 = o(σ 2 n 1 ,n 2 ).
Using stationarity and C 1 (σ n 1 ,n 2 )(a) and (2.6.6), we derive

B 2 ≤ ∥S n 1 ,ℓn 2 ∥ 2 E k i=2 S in 1 ,ℓn 2 -S (i-1)n 1 ,ℓn 2 |F n 1 ,ℓn 2 2 ≤ (k -1) ∥S n 1 ,ℓn 2 ∥ 2 ∥E (S n 1 ,ℓn 2 |F 0,ℓn 2 )∥ 2 = o(σ 2 n 1 ,n 2 ). Therefore B = o(σ 2 n 1 ,n 2
). Now we handle the quantity A and write

|A| ≤ k-1 i=2 k j=i+1 E (S n 1 ,ℓn 2 -E [S n 1 ,ℓn 2 |F n 1 ,ℓn 2 ]) S (j-i+1)n 1 ,ℓn 2 -S (j-i)n 1 ,ℓn 2 + k-1 i=2 k j=i+1 E (E [S n 1 ,ℓn 2 |F n 1 ,ℓn 2 ]) S (j-i+1)n 1 ,ℓn 2 -S (j-i)n 1 ,ℓn 2 := A 1 + A 2 .
Thereby, with similar arguments as above,

A 1 ≤ k 2 ∥S n 1 ,ℓn 2 ∥ 2 ∥S n 1 ,ℓn 2 -E (S n 1 ,ℓn 2 |F 0,ℓn 2 )∥ 2 = o(σ 2 n 1 ,n 2 ), A 2 ≤ k 2 ∥S n 1 ,ℓn 2 ∥ 2 ∥E (S n 1 ,ℓn 2 |F 0,ℓn 2 )∥ 2 = o(σ 2 n 1 ,n 2 ).
Hence A = o(σ 2 n 1 ,n 2 ). This ends the proof of (2.2.1). Now we prove (2.2.3). Let introduce the two following conditions :

lim x∧y→∞ k[x]+k+1 i=1 ℓ[y]+ℓ+1 j=1 X i,j -[kx] i=1 [ℓy] j=1 X i,j 2 σ [x],[y] = 0 (2.6.7)
and

lim x∧y→∞ k[x]+k+1 i=1 ℓ[y]+ℓ+1 j=1 X i,j -k[x] i=1 ℓ[y] j=1 X i,j 2 σ [x],[y] = 0. ( 2
a = √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=j P 0,0 (X h,k ) and A = n 1 n 2 j=1 n 1 h=-n 1 n 2 k=j P 0,0 (X h,k ) 2 2
.

We have

1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 s n 1 ,n 2 n 1 -i h=1-i n 2 -j k=1-j P 0,0 (X h,k ) -m 2 2 = 1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=-n 2 P 0,0 (X h,k ) -m -a -b -c -d 2 2 ≤ √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=-n 2 P 0,0 (X h,k ) -m 2 2 + A + B + C + D s 2 n 1 ,
√ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) -→ m in L 2 ,
(2.6.9)

n 1 n 2 j=1 n 1 h=0 n 2 k=n 2 -j+1 P 0,0 (X h,k ) 2 2 = o(s 2 n 1 ,n 2 ), (2.6.10
)

n 1 i=1 n 2 j=1 n 1 h=n 1 -i+1 n 2 -j k=0 P 0,0 (X h,k ) 2 2 = o(s 2 n 1 ,n 2 ).
(2.6.11) 40

Note that the following decomposition holds: for every

n 1 ≥ 1, n 2 > j > 0, √ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=n 2 -j+1 P 0,0 (X h,k ) = √ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) - n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) + n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) 1 - √ n 1 n 2 n 1 (n 2 -j) s n 1 ,n 2 -j s n 1 ,n 2 . Since s n 1 ,n 2 = √ n 1 n 2 h 1 (n 1 )h 2 (n 2 )
with h 1 and h 2 two slowly varying functions, we get

√ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=n 2 -j+1 P 0,0 (X h,k ) = √ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) - n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) + n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) 1 - h 2 (n 2 -j) h 2 (n 2 ) .
It follows that condition (2.6.9) implies condition (2.6.10) provided

1 n 2 n 2 j=1 1 - h 2 (j -1) h 2 (n 2 ) 2 -→ 0.
This holds since h 2 is a one-parameter slowly varying function. With similar arguments and using that h 1 is a one-parameter slowly varying function, we get that (2.6.9) implies (2.6.11).

2.6.4

Proof of Corollary 2.2.7

This proof is quite similar to its one dimensional version (see Corollary 1in [START_REF] Dedecker | On the weak invariance principle for nonadapted sequences under projective criteria[END_REF]). We give it for completeness.

Let rst prove (2.2.8) ⇒ C 1 ( √ n 1 n 2 ). Taking s n 1 ,n 2 = √ n 1 n 2 and m = i,j∈Z 2 P 0,0 (X i,j
), it follows that if the rst part of (2.2.8) holds then the conditions (2.2.4)-(2.2.6) of Proposition 2.2.6 are clearly satised. Therefore C 1 ( √ n 1 n 2 )(c) holds. Now starting from the decomposition (2.6.1) and using orthogonality, we derive

∥S n 1 ,n 2 ∥ 2 2 n 1 n 2 = 1 n 1 n 2 ∥S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]∥ 2 2 + 1 n 1 n 2 ∥R n 1 ,n 2 (S n 1 ,n 2 )∥ 2 2 + 1 n 1 n 2 ∥E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 )∥ 2 2 := A 2 + B 2 + C 2 , (2.6.12) where R n 1 ,n 2 (S n 1 ,n 2 ) = E [S n 1 ,n 2 |F n 1 ,0 ] + E [S n 1 ,n 2 |F 0,n 2 ] -E [S n 1 ,n 2 |F 0,0 ]. According to the proof of Theorem 2.2.2, since C 1 ( √ n 1 n 2 )(c) holds, C 2 converges to ∥m∥ 2 2 . Since by assumption, (n 1 n 2 ) -1 ∥S n 1 ,n 2 ∥ 2 2 converges to ∥m∥ 2 2 , we get overall that A and B converge to zero. Note that A → 0 is exactly C 1 ( √ n 1 n 2 )(b). It remains to prove that C 1 ( √ n 1 n 2 )(a) is satised. With this aim, note that since M n 1 ,n 2 := n 1 i=1 n 2
j=1 m • T i,j is an ortho-martingale with respect to ltration (F i,j ) i,j∈Z , we have

E (S n 1 ,n 2 |F n 1 ,0 ) = E [E (S n 1 ,n 2 |F n 1 ,n 2 ) -R n 1 ,n 2 (S n 1 ,n 2 ) + R n 1 ,n 2 (S n 1 ,n 2 ) -M n 1 ,n 2 |F n 1 ,0 ] . Therefore ∥E (S n 1 ,n 2 |F n 1 ,0 )∥ 2 √ n 1 n 2 ≤ B + 1 √ n 1 n 2 ∥E (S n 1 ,n 2 |F n 1 ,n 2 ) -R n 1 ,n 2 (S n 1 ,n 2 ) -M n 1 ,n 2 ∥ 2 ,
which converges to zero by C 1 ( √ n 1 n 2 )(c) and the fact that B → 0. Similarly, we get that (n

1 n 2 ) -1/2 ∥E (S n 1 ,n 2 |F 0,n 2 )∥ 2 → 0 and (n 1 n 2 ) -1/2 ∥E (S n 1 ,n 2 |F 0,0 )∥ 2 → 0. Hence C 1 ( √ n 1 n 2 )(a) holds.
It remains to prove that (2.2.9) ⇒ (2.2.8). Clearly if (2.2.9) holds so does the rst part of (2.2.8). Next we shall prove that

1 n 1 n 2 E S 2 n 1 ,n 2 → (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] a.s. and E m 2 = (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] ,
(2.6.13) which implies the second part of (2.2.8). Since X 0,0 is regular, the following decomposition holds

E [X 0,0 X k 1 ,k 2 ] = (i 1 ,i 2 )∈Z 2 (j 1 ,j 2 )∈Z 2 E [P i 1 ,i 2 (X 0,0 )P j 1 ,j 2 (X k 1 ,k 2 )] .
By orthogonality (which comes from the fact that the ltration (F i,j ) is commuting) and stationarity, we derive

E [X 0,0 X k 1 ,k 2 ] = (i 1 ,i 2 )∈Z 2 E [P 0,0 (X i 1 ,i 2 )P 0,0 (X k 1 +i 1 ,k 2 +i 2 )] . (2.6.14)
Hence

|E [X 0,0 X k 1 ,k 2 ]| ≤ (i 1 ,i 2 )∈Z 2 ∥P 0,0 (X i 1 ,i 2 )∥ 2 ∥P 0,0 (X k 1 +i 1 ,k 2 +i 2 )∥ 2 , so that (k 1 ,k 2 )∈Z 2 |E [X 0,0 X k 1 ,k 2 ]| ≤   (i 1 ,i 2 )∈Z 2 ∥P 0,0 (X i 1 ,i 2 )∥ 2   2 ,
which is nite under condition (2.2.9). Therefore, the series

(k 1 ,k 2 )∈Z 2 |E [X 0,0 X k 1 ,k 2 ]| converges.
In addition, note that with (2.6.14), we have

(k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] = (i 1 ,i 2 )∈Z 2 (j 1 ,j 2 )∈Z 2 E [P 0,0 (X i 1 ,i 2 )P 0,0 (X j 1 ,j 2 )] .
In addition,

E m 2 = (i 1 ,i 2 )∈Z 2 (j 1 ,j 2 )∈Z 2 E [P 0,0 (X i 1 ,i 2 )P 0,0 (X j 1 ,j 2 )] .
So,

E m 2 = (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] .
On another hand,

1 n 1 n 2 E S 2 n 1 ,n 2 = n 1 -1 i 1 =-n 1 +1 n 2 -1 i 2 =-n 2 +1 1 - |i 1 | n 1 1 - |i 2 | n 2 E (X 0,0 X i 1 ,i 2 ) , which then converges to (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] = E (m 2
). This ends the proof of (2.2.8). by max k,l , and max 1≤k≤n 1 by max k respectively for l. All along the proof, C will be a universal positive constant (depending on the dimension) which may vary from line to line. Our main aim is to prove that lim λ→∞ lim sup

n 1 ∧n 2 →∞ 1 s 2 n 1 ,n 2 E max k,l S 2 k,l 1 max k,l| S k,l| >λsn 1 ,n 2 = 0.
Note that the following decomposition holds: for every

1 ≤ k ≤ n 1 , 1 ≤ l ≤ n 2 , S k,l = S k,l -E [S k,l |F n 1 ,n 2 ] + R n 1 ,n 2 (S k,l ) + E [S k,l |F n 1 ,n 2 ] -R n 1 ,n 2 (S k,l ), (2.6.15) where R n 1 ,n 2 (S k,l ) = E [S k,l |F n 1 ,0 ]+E [S k,l |F 0,n 2 ]-E [S k,l |F 0,0 ]. Starting from (2.6.15)
and using the fact that for any A ≥ 0,

x 2 1 |x|>A ≤ 4 |x| - A 2 2 +
, where (x) + = x1 x≥0 , we get by convexity that, for any λ ≥ 0,

E max k,l S 2 k,l 1 max k,l| S k,l| >2λsn 1 ,n 2 ≤ 4E max k,l |S k,l | -λs n 1 ,n 2 2 + ≤ 12E max k,l |S k,l -E [S k,l |F n 1 ,n 2 ]| 2 + 12E max k,l |R n 1 ,n 2 (S k,l )| 2 + 12E max k,l |E [S k,l |F n 1 ,n 2 ] -R n 1 ,n 2 (S k,l )| -λs n 1 ,n 2 2 + .
Therefore by the condition C 2 (s n 1 ,n 2 )(a) and (b), it is sucient to show that,

lim λ→∞ lim n 1 ∧n 2 →∞ 1 s 2 n 1 ,n 2 E max k,l |E [S k,l |F n 1 ,n 2 ] -R n 1 ,n 2 (S k,l )| -λs n 1 ,n 2 2 + = 0.
(2.6.16)

For the sake of simplicity, we set S k,l :

= E [S k,l |F n 1 ,n 2 ] -R n 1 ,n 2 (S k,l ), S + n 1 ,n 2 := max(0, S 1,1 , . . . , S n 1 ,n 2 ), and S - n 1 ,n 2 := max(0, -S 1,1 , . . . , -S n 1 ,n 2 ).
Therefore if we assume that

C 2 (s n 1 ,n 2 )(a) and C 2 (s n 1 ,n 2 )(b) hold, then max k,l S 2 k,l s 2 n 1 ,n 2 (n 1 ,n 2 )∈(N * ) 2
will be uniformly integrable as soon as

lim λ→∞ lim sup n 1 ∧n 2 →∞ 1 s 2 n 1 ,n 2 E S + n 1 ,n 2 -λs n 1 ,n 2 2 +
= 0 (2.6.17) and lim λ→∞ lim sup

n 1 ∧n 2 →∞ 1 s 2 n 1 ,n 2 E S - n 1 ,n 2 -λs n 1 ,n 2 2 + = 0.
(2.6.18)

Note that using the projective operators, we have

S k,l = k u=1 l v=1 u-1 i=u-n 1 v-1 j=v-n 2 P u-i,v-j (X u,v ) = k-1 i=1-n 1 l-1 j=1-n 2 k∧(n 1 +i) u=1∨(i+1) l∧(n 2 +j) v=1∨(j+1) P u-i,v-j (X u,v ).
For any xed positive integer i and j, we introduce the double indexed sequence

(Y i,j,k,l ) k,l≥1 dened by Y i,j,k,l = k∧(n 1 +i) u=1∨(i+1) l∧(n 2 +j) v=1∨(j+1) P u-i,v-j (X u,v ).
Notice that, (Y i,j,k,l ) k,l≥1 is an ortho-martingale w.r.t the ltration (F i,j ) i,j∈Z .

With these notations,

S k,l = k-1 i=1-n 1 l-1 j=1-n 2 Y i,j,k,l . Setting b i,j = u i,j ( n 1 k=-n 1 n 2 ℓ=-n 2 u k,ℓ ) -1
, with (u i,j ) i,j∈Z a positive sequence such that

√ n 1 n 2 s n 1 ,n 2 n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 u i,j < ∞, we get S k,l -λs n 1 ,n 2 + ≤ k-1 i=1-n 1 l-1 j=1-n 2 (Y i,j,k,l -λb i,j s n 1 ,n 2 ) + .
(2.6.19) By Hölder's inequality and taking the maximum over (k, l) on both sides, we derive that

S + n 1 ,n 2 -λs n 1 ,n 2 2 + ≤ n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 u i,j n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 1 u i,j Y + i,j,n 1 ,n 2 -λb i,j s n 1 ,n 2 2 + ,
where Y + i,j,n 1 ,n 2 = max k,l (max(0, Y i,j,k,l )). Hence to prove (2.6.17), it is sucient to show that,

lim λ→∞ lim sup n 1 ∧n 2 →∞ 1 √ n 1 n 2 s n 1 ,n 2 n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 1 u i,j E Y + i,j,n 1 ,n 2 -λb i,j s n 1 ,n 2 2 + = 0.
(2.6.20)

To prove this, we will need the two following lemmas, whose proofs are postponed to Section 7.

Dénition 2.6.1. We say that a random eld

(M n 1 ,n 2 ) (n1,n 2 )∈N 2 is an ortho-submartingale w.r.t the commuting ltration (F n 1 ,n 2 ) (n 1 ,n 2 )∈Z 2 if M n 1 ,n 2 is integrable, F n 1 ,n 2 -measurable,
and for all integers i, j, k, ℓ, we have

E (M i,j |F k,ℓ ) ≥ M min(i,k),min(j,ℓ) . Lemma 2.6.2. Let M n 1 ,n 2 ≡ n 1 i=1 n 2
j=1 d i,j be an ortho-submartingale. Then for all p ∈]1, +∞[, and λ > 0

E max k,l |M k,l | p 1 max k,l| M k,l| >λ ≤ 2 2p p p -1 2 E |M n 1 ,n 2 | p 1 |Mn 1 ,n 2 |> λ 4 . Lemma 2.6.3. Let M n 1 ,n 2 ≡ n 1 i=1 n 2 j=1 d i,j be an ortho-martingale in L 2 . Then for all λ > 0 E (M n 1 ,n 2 -λ) 2 + ≤ n 1 i=1 E   n 2 j=1 d i,j 2
1 max k |Mk,n 2 |>λ   , (2.6.21)
and

E |M n 1 ,n 2 | 2 1 |Mn 1 ,n 2 |>λ ≤ 8 n 1 i=1 E   n 2 j=1 d i,j 2
1 max k |Mk,n 2 |> λ 2   .
(2.6.22)

In addition, for any xed i (the same goes for j), we have

E   n 2 j=1 d i,j -λ 2 +   ≤ n 2 j=1 E d 2 i,j 1 max l≤n 2 | l j=1 d i,j |>λ . (2.6.23)
To reduce the complexity of the notation, we write

n 1 -1 i=1-n 1 n 2 -1 j=1
-n 2 for i,j , and k∧(n 1 +i) u=1∨(i+1) by k u , the same goes for l v . Now we denote

U (n 1 , n 2 , λ) := 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E Y + i,j,n 1 ,n 2 -λb i,j s n 1 ,n 2 2 + . Since (|x| -λ) 2 + ≤ x 2 1 |x|>λ , we have U (n 1 , n 2 , λ) ≤ 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E Y + i,j,n 1 ,n 2 2 1 Y + i,j,n 1 ,n 2 >λb i,j sn 1 ,n 2 .
Next applying Lemma 2.6.2, we get

U (n 1 , n 2 , λ) ≤ C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E Y 2 i,j,n 1 ,n 2 1 |Yi,j,n 1 ,n 2 |> λ 4 b i,j sn 1 ,n 2 .
Then using (2.6.22) of Lemma 2.6.3, we derive

U (n 1 , n 2 , λ) ≤ C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u E   n 2 v P u-i,v-j (X u,v ) 2
1 maxu|Y i,j,u,n 2 |> λ 8 b i,j sn 1 ,n 2   .
Let A > 0 and set A i,j,n 2 = Au 2 i,j n 2 , we have that U (n 1 , n 2 , λ) is less than or equal to

C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u E   n 2 v P u-i,v-j (X u,v ) 2 1 ( n 2 v P u-i,v-j (Xu,v)) 2 >A i,j,n 2   + C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 A i,j,n 2 P max u |Y i,j,u,n 2 | > λ 8 b i,j s n 1 ,n 2 := C(I + II).
Using (2.6.23) of Lemma 2.6.3, we get

I ≤ C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v E P 2 u-i,v-j (X u,v )1 max l| l v P u-i,v-j (Xu,v)|> √ A i,j,n 2 2
.

It follows that for any B > 0

I ≤ C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v E P 2 u-i,v-j (X u,v )1 P 2 u-i,v-j (Xu,v)>Bu 2 i,j + C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j P max l l v P u-i,v-j (X u,v ) > A i,j,n 2 2 .
By stationarity, it follows that,

I ≤ C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E P 2 0,0 (X i,j )1 P 2 0,0 (X i,j )>Bu 2 i,j + C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j P max l l v P u-i,v-j (X u,v ) > A i,j,n 2 2 . := C 1 (I 1 + I 2 ).
The quantity I 1 converges to zero by C 2 (s n 1 ,n 2 )(c), if we rst let n 1 ,n 2 tend to innity and then B to innity. To handle I 2 , we use Doob's inequality that leads to

I 2 ≤ 4 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j n 2 ∧(n 2 +j) v=1∨(j+1) E [P u-i,v-j (X u,v )] 2 A i,j,n 2 .
Using stationarity again, and recalling that A i,j,n 2 = Au 2 i,j n 2 , we derive

I 2 ≤ 4 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j n 2 E P 2 0,0 (X i,j ) Au 2 i,j n 2 ≤ 4 B A √ n 1 n 2 s n 1 ,n 2 i,j E P 2 0,0 (X i,j ) u i,j . Now by C 2 (s n 1 ,n 2 )(c), we have sup n 1 ≥1 n 2 ≥1 √ n 1 n 2 s n 1 ,n 2 n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 E P 2 0,0 (X i,j ) u i,j < ∞. (2.6.24)
Hence I 2 converges to zero if we rst let A goes to innity and then B to innity. So overall I converges to zero by letting rst n 1 ∧ n 2 tend to innity and then A. Now to deal with II, we proceed as for I 2 . Indeed, Doob's maximal inequality leads to

II = A √ n 1 n 2 s n 1 ,n 2 i,j u i,j P max u |Y i,j,u,n 2 | > λ 8 b i,j s n 1 ,n 2 , ≤ C 2 A √ n 1 n 2 s n 1 ,n 2 i,j u i,j E Y 2 i,j,n 1 ,n 2 λ 2 b 2 i,j s 2 n 1 ,n 2 .
Hence, by stationarity,

II ≤ C 2 A λ 2 √ n 1 n 2 s n 1 ,n 2 i,j E P 2 0,0 (X i,j ) u i,j √ n 1 n 2 s n 1 ,n 2 i,j u i,j 2
, which converges to zero by letting rst n 1 ∧ n 2 tend to innity and then λ. This ends the proof of (2.6.17). The proof of (2.6.18) can be done similarly.

2.6.6

Proof of Proposition 2.3.3

Notice rst that the condition C 2 (s n 1 ,n 2 )(a) holds as soon as,

sup 1≤k≤n 1 1≤l≤n 2 |E [S k,l |F n 1 ,0 ] -E [S k,l |F 0,0 ]| 2 = o(s n 1 ,n 2 ), (2.6.25
)

sup 1≤k≤n 1 1≤l≤n 2 |E [S k,l |F 0,n 2 ] -E [S k,l |F 0,0 ]| 2 = o(s n 1 ,n 2 ), (2.6.26)
and

sup 1≤k≤n 1 1≤l≤n 2 |E [S k,l |F 0,0 ]| 2 = o(s n 1 ,n 2 ).
(2.6.27)

Note that

sup 1≤k≤n 1 1≤l≤n 2 |E [S k,l |F 0,0 ]| 2 ≤ n 1 k=1 n 2 ℓ=1 ∥E (X k,ℓ |F 0,0 )∥ 2 ,
and since

X k,ℓ = +∞ i=-∞ +∞ j=-∞ P 0,0 (X k,ℓ ), E (X k,ℓ |F 0,0 ) = 0 i=-∞ 0 j=-∞ P i,j (X k,ℓ ).
Therefore, the condition (2.6.27) is implied by C 3 (s n 1 ,n 2 )(a). Now, we focus on (2.6.25). Notice rst that

E [S k,l |F n 1 ,0 ] -E [S k,l |F 0,0 ] = k u=1 ℓ v=1 n 1 i=1 0 j=-∞ P i,j (X u,v ).
Thus (2.6.25) is equivalent to

sup 1≤k≤n 1 1≤l≤n 2 k u=1 ℓ v=1 n 1 i=1 0 j=-∞ P i,j (X u,v ) 2 = o(s n 1 ,n 2 ). (2.6.28) But sup 1≤k≤n 1 1≤l≤n 2 k u=1 ℓ v=1 n 1 i=1 0 j=-∞ P i,j (X u,v ) 2 ≤ n 2 v=1 sup 1≤k≤n 1 k u=1 n 1 i=1 0 j=-∞ P i,j (X u,v ) 2 ≤ n 2 v=1 sup 1≤k≤n 1 k u=1 u-1 i=u-n 1 0 j=-∞ P u-i,j (X u,v ) 2 ≤ n 2 v=1 sup 1≤k≤n 1 k-1 i=1-n 1 k∧(i+n 1 ) u=(i+1)∨1 0 j=-∞ P u-i,j (X u,v ) 2 .
We denote

0 j=-∞ P u-i,j (X u,v ) by D i,u (v), and notice that D i,u (v) is orthogonal with respect to u. Next denote k∧(i+n 1 ) u=(i+1)∨1 D i,u (v) by Y i,k (v). Let (U i (v)) i≥1-n 1
be a sequence of positive real numbers that we will dene later. By Cauchy-Schwarz's inequality,

k-1 i=1-n 1 Y i,k (v) 2 = k-1 i=1-n 1 U i (v) 1 U i (v) Y i,k (v) 2 ≤ k-1 i=1-n 1 U i (v) k-1 i=1-n 1 1 U i (v) (Y i,k (v)) 2 .
Therefore,

sup 1≤k≤n 1 k-1 i=1-n 1 Y i,k (v) 2 2 ≤ n 1 -1 i=1-n 1 U i (v) n 1 -1 i=1-n 1 1 U i (v) sup 1≤k≤n 1 |Y i,k (v)| 2 2
.

52

Note now that (Y i,k (v)) k≥1 is a martingale. Using Doob's inequality and stationarity, it follows that

sup 1≤k≤n 1 |Y i,k (v)| 2 2 ≤ 4 n 1 ∧i+n 1 u=(i+1)∨1 ∥D i,u (v)∥ 2 2 ≤ 4 n 1 ∧i+n 1 u=(i+1)∨1 0 j=-∞ ∥P u-i,j (X u,v )∥ 2 2 ≤ 4n 1 +∞ j=v ∥P 0,0 (X i,j )∥ 2 2 .
So overall taking U i (v) = +∞ j=v ∥P 0,0 (X i,j )∥ 2 2 , we have

sup 1≤k≤n 1 k-1 i=1-n 1 Y i,k (v) 2 2 ≤ 4n 1   n 1 -1 i=1-n 1 +∞ j=v ∥P 0,0 (X i,j )∥ 2 2   2 .
Therefore

n 2 v=1 sup 1≤k≤n 1 k-1 i=1-n 1 Y i,k (v) 2 ≤ 2 √ n 1 n 1 -1 i=1-n 1 n 2 v=1 +∞ j=v ∥P 0,0 (X i,j )∥ 2 2 .
Thus

S k,ℓ -E (S k,ℓ |F n 1 ,n 2 ) = k u=1 ℓ v=1 n 1 i=-∞ +∞ j=n 2 +1 P i,j (X u,v ) + +∞ i=n 1 +1 n 2 j=-∞ P i,j (X u,v ) + +∞ i=n 1 +1 +∞ j=n 2 +1 P i,j (X u,v ) .
Hence C 2 (s n 1 ,n 2 )(b) holds as soon as,

sup 1≤k≤n 1 1≤l≤n 2 k u=1 ℓ v=1 n 1 i=-∞ +∞ j=n 2 +1 P i,j (X u,v ) 2 = o(s n 1 ,n 2 ), (2.6.29
)

sup 1≤k≤n 1 1≤l≤n 2 k u=1 ℓ v=1 +∞ i=n 1 +1 n 2 j=-∞ P i,j (X u,v ) 2 = o(s n 1 ,n 2 ), (2.6
n 1 i=-∞ +∞ j=n 2 +1 P i,j (X u,v ) = n 1 i=1 +∞ j=n 2 +1 P i,j (X u,v ) + 0 i=-∞ +∞ j=n 2 +1 P i,j (X u,v ).
Therefore, (2.6.29) holds as soon as We now prove that in adapted case C 3 (s n 1 ,n 2 ) holds as soon as C 3 (s n 1 ,n 2 ) does.

sup 1≤k≤n 1 1≤l≤n 2 k u=1 ℓ v=1 n 1 i=1 +∞ j=n 2 +1 P i,j (X u,v ) 2 = o(s n 1 ,n 2 ), (2
In adapted case C 3 (s n 1 ,n 2 )(a) reads as

n 1 u=1 n 2 v=1 ∥E (X u,v |F 0,0 )∥ 2 = o(s n 1 ,n 2 ),
which is implied by C 3 (s n 1 ,n 2 )(a). Next, note that C 3 (s n 1 ,n 2 )(b) can be rewritten as

√ n 1 n 1 i=1 n 2 v=1 P -ĩ,-v (X 0,0 ) 2 = o(s n 1 ,n 2 ),
where P ĩ,j (X) = E (X|F i,j ) -E (X|F i-1,j ). We then apply Lemma 6.1 in [START_REF] Dedecker | Invariance principles for linear processes with application to isotonic regression[END_REF] to the partial sum with b i = 1 and u i = P -ĩ,-v (X 0,0 ) 2 . Hence we get that

n 1 i=1 n 2 v=1 P -ĩ,-v (X 0,0 ) 2 ≤ C n 1 i=1 1 √ i n 2 v=1 n 1 k=i P -k,-v (X 0,0 ) 2 2 1/2
, Therefore, we deduce that

n 1 i=1 n 2 v=1 P -ĩ,-v (X 0,0 ) 2 ≤ C n 1 i=1 n 2 v=1 1 √ i ∥E (X i,v |F 0,0 )∥ 2 .
Hence, C 3 (s n 1 ,n 2 )(b) is implied by C 3 (s n 1 ,n 2 )(a). Similar arguments are applied to

C 3 (s n 1 ,n 2 )(c) (implied by C 3 (s n 1 ,n 2 )(b)). 2.6.7 Proof of Theorem 2.3.5 Since C 2 (s n 1 ,n 2 ) holds, the sequence s -2 n 1 ,n 2 max 1≤k≤n 1 1≤l≤n 2 S 2 k,l (n 1 ,n 2 )∈N 2
is uniformly integrable, and the process s -1

n 1 ,n 2 S [n 1 t 1 ],[n 2 t 2 ] (t 1 ,t 2 )∈[0,1] 2 is tight.
It remains to prove that the nite-dimensional distributions converge, that is, for any t (1)

= (t 1 1 , t 1 2 ), . . . , t (k) = (t k 1 , t k 2 ) ∈ [0, 1] 2 , 1 s n 1 ,n 2   [n 1 t (1) 1 ] k=1 [n 2 t (1) 2 ] l=1 X k,l , . . . , [n 1 t (k) 1 ] k=1 [n 2 t (k) 2 ] l=1 X k,l   ⇒ σ W 2 (t (1) ), . . . , W 2 (t (k) ) .
ing Corollary 2.10 in [START_REF] Merlevède | Functional Gaussian Approximation for Dependent Structures[END_REF] to the submartingale (max

1≤k≤n 1 M k,n 2 ) n 2 ∈N we have E max 1≤l≤n 2 max 1≤k≤n 1 |M k,l | p 1 max 1≤l≤n 2 max 1≤k≤n 1 |M k,l |>λ ≤ 2 p p p -1 E max 1≤k≤n 1 |M k,n 2 | p 1 max 1≤k≤n 1 |Mk,n 2 |> λ 2 .
The desired result follows by applying Corollary 2.10 in [START_REF] Merlevède | Functional Gaussian Approximation for Dependent Structures[END_REF] again to (M k,n 2 ) k∈N . . Finally (2.6.23) is again an application of inequality (3.6)] in [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF].

57

Chapter 3

On the central limit theorem for stationary random elds under

L 1 -projective condition
The rst aim of this chapter is to wonder to what extent we can generalize the central limit theorem of Gordin [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF] under the so-called L 1 -projective criteria to ergodic stationary random elds when completely commuting ltrations are considered. Surprisingly it appears that this result cannot be extended to its full generality and that an additional condition is needed.

Introduction and main results

Let (Ω, A, P) be a probability space, and T : Ω → Ω be an ergodic bijective bimeasurable transformation preserving the probability P. Let F 0 be a sub-σ-algebra of

A satisfying F 0 ⊆ T -1 (F 0 )
and f be a L 1 (P) real-valued centered random variable adapted to F 0 . By U we denote the operator U : f → f • T . The notation I will denote the identity operator. Dene then the stationary sequence (f i ) i∈Z by

f i = f • T i = U i f , its associated stationary ltration (F i ) i∈Z by F i = F 0 • T -i and let S n (f ) = n-1 i=0 U i f .
The following theorem is essentially due to Gordin [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF] and gives sucient conditions for (U i f ) i∈Z to satisfy the central limit theorem.

Theorem 3.1.1 (Gordin). Assume that the series

i≥0 E(U i f |F 0 ) converges in L 1 (P) (3.1.1)
and

lim inf n→∞ E(|S n (f )|) √ n < ∞ . (3.1.2)
Then n -1/2 S n (f ) converges in distribution to a centered normal variable (that can be degenerate).

The proof of this result is based on the following coboundary martingale decomposition (see [START_REF] Volný | Approximating martingales and the central limit theorem for strictly stationary processes[END_REF] for more details concerning necessary and sucient conditions for the existence of such a decomposition): Under (3.1.1),

f = m + (I -U )g (3.1.3)
where m and g are in L 1 (P) and (U i m) i≥0 is a stationary sequence of martingale dierences, and on the following theorem whose complete proof can be found in Esseen and Janson [START_REF] Esseen | On moment conditions for normed sums of independent variables and martingale dierences[END_REF].

Theorem 3.1.2 (Esseen-Janson). If (U i m) i≥0 is a stationary and ergodic sequence of martingale dierences in L 1 (P) satisfying

lim inf n→∞ E(| n-1 i=0 U i m|) √ n < ∞ , (3.1.4)
then m ∈ L 2 (P).

Clearly, using the coboundary martingale decomposition The aim of this paper is to prove that Theorem 3.1.1 can be extended to random elds when the underlying ltrations are completely commuting (see [START_REF] Khoshnevisan | Multiparameter processes. An introduction to random elds[END_REF]Chap. 1] for a denition of this notion). To x the idea, let us rst state the result in case of multidimensional index of dimension d = 2 (the general case will be stated in Section 3.4). Then, in complement to the previous notation, let S be an ergodic bimeasurable and measure preserving bijection of Ω. By V we denote the operator

V : f → f • S.
In what follows we shall assume that the ergodic transformations T and S are commuting. Note that T i,j = T i S j is an ergodic Z 2 action on (Ω, A, P). Let F 0,0 be a sub-sigma eld of A and for all (i, j) ∈ Z 2 dene F i,j = T -i S -j (F 0,0 ). Suppose that the ltration (F i,j ) (i,j)∈Z 2 is increasing in i for every j xed and increasing in j for every i xed, and is completely commuting in the sense that, for any integrable f ,

E(E(f |F i,j )|F u,v ) = E(f |F i∧u,j∧v ) .
To extend Theorem 3.1.1 to random elds indexed by the lattice Z 2 , the rst tool is a suitable coboundary orthomartingale decomposition. In what follows, f is a F 0,0 -measurable centered L 1 (P) function. According to Volný [START_REF] Volný | Martingale-coboundary representation for stationary random elds[END_REF], the condition:

the series i,j≥0 E(U i V j f |F 0,0 ) converges in L 1 (P) (3.1.5)
implies the existence of the following decomposition:

f = m + (I -U )g 1 + (I -V )g 2 + (I -U )(I -V )g 3 , (3.1.6)
where m, g 1 , g 2 , g 3 ∈ L 1 (P), (U i V j m) is a stationary eld of orthomartingale dierences, (V j g 1 ) j is a stationary martingale dierences sequence with respect to the ltration (F ∞,j ) j , and (U i g 2 ) i is a stationary martingale dierences sequence with respect to the ltration

(F i,∞ ) i . To x the ideas, setting E a,b (•) = E(•|F a,b ), we have m = i,j≥0 E 0,0 (U i V j f ) -E -1,0 (U i V j f ) -E 0,-1 (U i V j f ) + E -1,-1 (U i V j f ) , g 1 = i,j≥0 E -1,0 (U i V j f )-E -1,-1 (U i V j f ) , g 2 = i,j≥0 E 0,-1 (U i V j f )-E -1,-1 (U i V j f ) ,
and

g 3 = i,j≥0 E -1,-1 (U i V j f ).
Recall also that (U i V j m) is said to be a orthomartingale dierences eld w.r.t. (F i,j ) if

E i-1,j (U i V j m) = E i,j-1 (U i V j m) = E i-1,j-1 (U i V j m) = 0 a.s.
Note that if f is additionally assumed to be regular in the sense that f is F ∞,∞measurable and E(f |F 0,-∞ ) = E(f |F -∞,0 ) = 0 then, it is proved in Volný [START_REF] Volný | Martingale-coboundary representation for stationary random elds[END_REF] that the converse is true, meaning that if f satises the decomposition (3.1.6) then (3.1.5) holds. We also refer to [START_REF] El Machkouri | Orthomartingale-coboundary decomposition for stationary random elds[END_REF] where the existence of the decomposition (3.1.6) is proved under a reinforcement of (3.1.5) (they assume that the series of the L 1 -norm is convergent). We also mention [24, Theorem 2.2] where a necessary and sucient condition for an orthomartingale-coboundary decomposition is established when all the underlying random elements are square integrable.

Our rst result is the following: Theorem 3.1.3. Let f be a F 0,0 -measurable centered L 1 (P) random variable. Let

S n 1 ,n 2 (f ) = n 1 i=0 n 2 j=0 U i V j f . Assume that condition (3.1.5) is satised and that lim inf n→∞ 1 √ n n-1 i=0 U i f 1 < ∞ , lim inf N →∞ 1 √ N N -1 j=0 V j f 1 < ∞ (3.1.7)
and lim inf

n 1 →∞ lim inf n 2 →∞ E(|S n 1 ,n 2 (f )|) √ n 1 n 2 < ∞ , lim inf n 2 →∞ lim inf n 1 →∞ E(|S n 1 ,n 2 (f )|) √ n 1 n 2 < ∞ . (3.1.8)
Then the random variables m, (I -U )g 1 and (I -V )g 2 dened in (

(n 1 n 2 ) -1/2 S n 1 ,n 2 (m)
to have the same limiting behavior (to see this it suces to take 2-p) in the construction of the counterexample of Theorem 3.1.4).

n k = [2 k/2 ] and m k ∼ (n k /k) p/(
However, reinforcing the conditions of Theorem 3.1.3, we can prove the following CLT.

Theorem 3.1.5. In addition to the conditions of Theorem 3.1.3, assume that

lim min(n 1 ,n 2 )→∞ E(|S n 1 ,n 2 (f )|) √ n 1 n 2
exists.

(3.1.9)

Then, as

min(n 1 , n 2 ) → ∞, (n 1 n 2 ) -1/2 S
1 √ nk ∥S n,k (f )∥ 1 ≤ lim n→∞ 1 n ∥S n,n (f )∥ 1 and lim sup k→∞ lim sup n→∞ 1 √ nk ∥S n,k (f )∥ 1 ≤ lim n→∞ 1 n ∥S n,n (f )∥ 1 .
This is consequence of the proof of Theorem 3.1.5. The existence of lim n→∞ n -1 ∥S n,n (f )∥ 1 has been mentioned there.

It is noteworthy to indicate that f does not need to be in L 2 but only in L 1 to apply Theorem 3.1.5 (see Example 5.1.10 given below). Theorem 3.1.5 then gives alternative projective conditions compared to those required in [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF]Th. 5.1] or in [27, Th. 1] for the central limit theorem under the normalization √ n 1 n 2 to hold. Note that the proofs of the two above mentioned results are also based on an orthomartingale approximation. We refer also to [START_REF] Wang | A new condition for the invariance principle for stationary random elds[END_REF] where the notion of orthomartingales and completely commuting ltrations have been previously used in the particular case of functions of iid random elds. Let us also indicate that when ltrations in the lexicographic order rather than completely commuting ltrations are considered, [21, Th.

1] provides a projective type condition in the spirit of the L 1 -projective condition (3.1.5) (but still requiring f to be in L 2 ) for the normalized partial sums associated with a stationary random eld to satisfy the central limit theorem. His proof is based on the so-called Lindeberg method.

Examples

3.2.1

An example when f is in

L 1 but not in L 2
For k ∈ N * and i, j ∈ Z, let e k,i,j be mutually independent zero mean random variables with U e k,i,j = e k,i+1,j , V e k,i,j = e k,i,j+1 . Let

F a,b = σ(e k,i,j , k ∈ Z, i ≤ a, j ≤ b).
We denote e k = e k,0,0 . Let (v k ) k≥1 be a sequence of nonnegative reals and (p k ) k≥1 a sequence of reals in [0, 1]. Assume that for any i, j ∈ Z 2 , L(e k,i,j ) = L(e k ) and that e k takes value v k with probability p k , -v k with probability p k and 0 with probability

1 -2p k . It follows that ∥e k ∥ 1 = 2v k p k and ∥e k ∥ 2 2 = 2v 2 k p k .
We use the following selection of (v k ) k≥1 and (p k ) k≥1 : v k = k 2 (log(k + 1)) 2 and p k = 1 2k 2 (log(k+1)) 4 . For any k ≥ 1 and i ≥ 0, let a k,i = (k + i) -2 , and dene

g = k≥1 i≥0 a k,i U -k-i e k , h = k≥1 1 k U -1 V -1 e k , m = k≥1 1 k 2 e k ,
and f = m + (I -U )g + (I -U )(I -V )h. By simple computations we have ∥g∥ 1 < ∞ and ∥h∥ 1 < ∞ but ∥g∥ 2 = ∞ and ∥h∥ 2 = ∞. In addition m is in L 2 (P). On another hand, f is in L 1 (P) but not in L 2 (P), and one can verify that i,j≥0 ∥E(U i V j f |F 0,0 )∥ 1 < ∞. Moreover, for any positive integer ℓ, by independence of the r.v.'s e k,i,j , we infer that

∥(I -U ℓ )g∥ 2 2 = k≥1 ℓ-1 i=0 a 2 k,i ∥e k ∥ 2 2 + k≥1 i≥0 (a k,i -a k,i+ℓ ) 2 ∥e k ∥ 2 2 .
Hence, by simple algebra, there exists a positive constant C such that ∥(I -U ℓ )g∥ 2 2 ≤ C log(ℓ + 1) for any positive integer ℓ.

(3.2.1)

In particular, we get ∥(I -U )g∥ 2 < ∞. In addition, by (3.2.1),

∥ n-1 i=0 N -1 j=0 U i V j (I -U )g∥ 2 2 nN = ∥(I -U n )g∥ 2 2 n → n→∞ 0 , which combined with ∥ n-1 i=0 N -1 j=0 U i V j (I-U )(I-V )h∥ 1 √ nN → 0, as max(n, N ) → ∞, implies that ∥S n,N (f -m)∥ 1 √ nN → 0, as min(n, N ) → ∞. Next, since, lim min(n,N )→∞ ∥ n-1 i=0 N -1 j=0 U i V j m∥ 1 √ nN
exists (it is equal to 2 π ∥m∥ 2 ), we can deduce that lim min(n,N )→∞

∥ n-1 i=0 N -1 j=0 U i V j f ∥ 1 √ nN
also exists. Hence all the conditions of Theorem 3.1.5 are satised. So, as min(n

1 , n 2 ) → ∞, (n 1 n 2 ) -1/2 S n 1 ,n 2 (f )
converges in distribution to a centered normal variable.

3.2.2

An example where f does not satisfy Hannan's L 2 -condition

We exhibit an example where f satises all the conditions of Theorem 3.1.5 but not the Hannan's L 2 -condition required in [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF]Th. 5.1]. We consider the random eld (e k,i,j ) k,i,j of mutually independent random variables as in Example 5.1.10 with the following choices of (v k ) k≥1 and (p k ) k≥1 . Let α > 4. Then for any k ≥ 1, dene

v k = k α and p k = 1 2k 5 log(k + 1) 2 .
For any k ≥ 1 and i, j ≥ 0, let a k,i,j = (k + i + j) -α . Then, dene

f = k≥1 u,v≥0 a k,u,v U -u V -v e k . (U i V j f ) i,j is usually called a super linear random eld. Let F a,b = σ(e k,i,j , k ∈ Z, i ≤ a, j ≤ b).
Note that f is a F 0,0 -measurable random variable, centered and in L 2 (P). In addition, one can check that condition (3.1.5) is satised implying that the orthomartingale coboundary decomposition (3.1.6) holds. Moreover, by simple algebra, one can verify that m ∈ L 2 (P) and that there exists a positive constant

K (depending on α) such that ∥(I -U ℓ )g 1 ∥ 2 2 + ∥(I -U ℓ )g 2 ∥ 2
2 ≤ Kℓ(log(ℓ + 1)) -1 . Proceeding as in Example 5.1.10, one can verify that conditions (3.1.7), (3.1.8) and (3.1.9) are satised. Hence, Theorem 3.1.5 applies to (n 1 n 2 ) -1/2 S n 1 ,n 2 (f ).

On another hand, dening P 0,0 (

•) = E 0,0 (•) -E -1,0 (•) -E 0,-1 (•) + E -1,-1 (•), we get, for any i, j ≥ 0, ∥P 0,0 (U i V j f )∥ 2 2 = k≥1 a k,i,j e k 2 2 = k≥1 a 2 k,i,j ∥e k ∥ 2 2 ≥ k≥i+j+1 k 2α-5 (k + i + j) 2α (log(k + 1)) 2 , implying that ∥P 0,0 (U i V j f )∥ 2 2 ≥ C (i + j + 1) 4 (log(i + j + 2)) 2 .
Hence i,j ∥P 0,0 (U i V j f )∥ 2 diverges. So the Hannan's L 2 -condition in the random elds setting does not hold, and [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF]Th. 5.1] does not apply. Note also that for this example, [21, Th. 1] that involves ltrations in the lexicographic order, cannot be applied.

Proofs

m ′ = m + (I -V )g 2 . (3.3.1) It follows that (U i m ′ ) i is a stationary sequence of L 1 (P) martingale dierences with respect to (F i,∞ ) i . Since T is ergodic, according to Theorem 3.1.2, if lim inf n→∞ E(| n-1 i=0 U i m ′ |) √ n < ∞ , (3.3.2)
then m ′ ∈ L 2 (P). By (3.1.6),

n-1 i=0 U i m ′ 1 ≤ n-1 i=0 U i f 1 + 2∥g 1 ∥ 1 + 4∥g 3 ∥ 1 .
Hence, since g 1 and g 3 are in L 1 (P), under the rst part of (3.1.7)

lim inf n→∞ E(| n-1 i=0 U i m ′ |) √ n ≤ lim inf n→∞ E(| n-1 i=0 U i f |) √ n < ∞ , Therefore (3.3.
2) holds and m ′ ∈ L 2 (P). Next recall that m ′ = m + (I -V )g 2 and recall that (V j m) j is a stationary sequence of L 1 (P) martingale dierences with respect to (F ∞,j ) j . Since S is ergodic, according again to Theorem 3.1.2, to prove that m ∈ L 2 (P), it suces to prove that

lim inf n→∞ E(| n-1 j=0 V j m|) √ n < ∞ . (3.3.3) But since m ′ = m + (I -V )g 2 and g 2 is in L 1 (P), proving (3.3.3) is reduced to show that lim inf n→∞ E(| n-1 j=0 V j m ′ |) √ n < ∞ . (3.3.4)
With this aim, recall rst that m ′ ∈ L 2 (P) and note that

1 √ n n-1 j=0 V j m ′ 1 ≤ 1 √ n n-1 j=0 V j m ′ 2 := σ n . (3.3.5)
For any xed positive integer n, let d := n -1/2 n-1 j=0 V j m ′ . Since (U i m ′ ) i is a stationary and ergodic sequence of martingale dierences in L 2 (P), so is (U i d) i . By the CLT for stationary and ergodic martingales in L 2 (P), as N → ∞, N -1/2 N -1 i=0 U i d converges in distribution to a centered Gaussian random variable G n with variance σ 2 n . Hence, by [START_REF] Billingsley | Convergence of probability measures[END_REF]Th. 3.4] and noticing that E|G n | = σ n 2/π, for any xed positive integer n, we get

σ n ≤ π 2 lim inf N →∞ 1 √ N N -1 i=0 U i d 1 = π 2 lim inf N →∞ 1 √ nN N -1 i=0 n-1 j=0 U i V j m ′ 1 . (3.3.6)
But, recalling (3.1.6) and that m ′ = m + (I -V )g 2 , we derive

1 √ N N -1 i=0 n-1 j=0 U i V j m ′ 1 ≤ 1 √ N N -1 i=0 n-1 j=0 U i V j f 1 + 2n √ N ∥g 1 ∥ 1 + 4 √ N ∥g 3 ∥ 1 .
Since g 1 and g 3 are in L 1 (P), the two last terms of the right-hand side are converging to zero as N tends to innity. Hence, taking into account (3.3.5) and (3.3.6), we get

lim inf n→∞ E(| n-1 j=0 V j m ′ |) √ n ≤ π 2 lim inf n→∞ lim inf N →∞ 1 √ nN N -1 i=0 n-1 j=0 U i V j f 1 .
which is nite by the second part of condition (3.1.8). This ends the proof of (3.3.4)

(and then of (3.3.3)) and leads to the fact that m is in L 2 (P). Next recall that we have proved that m ′ dened in (3.3.1) is in L 2 (P) which combined with the fact that m is in L 2 (P) implies that (I -V )g 2 is in L 2 (P). On another hand setting m ′′ = m + (I -U )g 1 , we can use previous arguments to infer that m ′′ is in L 2 (P).

Hence taking into account that m ∈ L 2 (P), we get that (I -U )g 1 is in L 2 (P). This ends the proof of the theorem. Let (X i,j ) (i,j)∈Z 2 be an iid random eld on (Ω, B, µ). Let A = σ{X i,j , (i, j) ∈ Z 2 } . Then, there exist transformations T and S such that X i,j • T = X i+1,j and X i,j • S = X i,j+1 . These transformations on (Ω, A, µ) are bijective, commuting, probability preserving and ergodic. Let U f = f • T and V f = f • S. Denote e = X 0,0 and

C = σ{U i e : i ∈ Z}. (C, T
) is thus a Bernoulli dynamical system and the sigma algebras S j C, j ∈ Z, are mutually independent. Let us recall the so-called Rokhlin lemma.

Lemma 3.3.1 (Rokhlin lemma). Let (Ω, A, µ, T ) be an ergodic dynamical system, n a positive integer, and ϵ > 0. Then there exists a set

F ∈ A such that F, T -1 F, . . . , T -n+1 F are disjoint and µ(∪ n-1 i=0 T -i F ) > 1 -ϵ.
For any integer k ≥ 1, we set n k = [2 k/2 ] and m k = 2 k . Let ϵ > 0. By Lemma 3.3.1 there exists a Rokhlin tower

F, T -1 F, . . . , T -N k +1 F with N k = m k n k ∼ 2 3k/2 . Note that 1/N k ≥ µ(T -i F ) > (1 -ϵ)/N k , for any i = 0, . . . , N k -1. We dene g k (ω) =          (j + 1) m k /n k on T -j F, j = 0 . . . , n k -1, (2n k -j -1) m k /n k on T -j F, j = n k , . . . , 2n k -1, 0 on the rest of Ω.
We can notice that

g k -U g k is equal to m k /n k on T -j F , j = 0 . . . , n k -1, to -m k /n k on T -j F , j = n k . . . , 2n k -1
, and to 0 on the rest of Ω. In addition, we have

∥g k -U g k ∥ 2 2 ≤ 2 n k m k ∼ 2 2 k/2 , ∥g k ∥ 1 ≤ 2 n k m k ≤ 2 1-k/4 and 2n k -1 i=0 U i (g k -U g k ) 2 = ∥g k -U 2n k g k ∥ 2 ≤ 2 √ n k .
Notice that all sums of U i (g k -U g k ) are C-measurable hence the random variables

(V j (g k -U 2n k g k )) j are iid. In addition, the support of g k -U 2n k g k is included in the union B k of F, . . . , T -4n k +1 F hence is of measure ≤ 4/m k = 2 -k+2 . Next, for any k ≥ 1, let A k = {|g k -U 2n k g k | ≥ (1/2) √ m k n k }. The set A k is included in B k and is of measure 2/m k = 2 -k+1
. Because the sigma algebras S i C, i ∈ Z, are mutually independent, the sets S -j A k , j = 0, . . . , m k -1, are independent. For h = 1 A k using that e 2 ln(1-x)/x ≥ e -4 for any x ∈]0, 1/2] and that m k µ(A k ) = 2, we thus have, for any k ≥ 2,

µ m k -1 j=0 V j h = 1 = m k µ(A k )(1 -µ(A k )) m k -1 ≥ 2/e 4 .
We then conclude that

µ 1 √ n k 1 √ m k 2n k -1 i=0 m k -1 j=0 U i V j (g k -U g k ) ≥ 1/2 ≥ 2/e 4 . ( 3

.3.7)

By recursion we shall dene a strictly increasing sequence k ℓ ↗ ∞ and then set

g = ∞ ℓ=1
g k ℓ and f = g -U g .

For ℓ = 1 we put k ℓ = 1. Suppose now that for 1 ≤ ℓ ′ < ℓ the k ℓ ′ have been dened. All the functions g k ℓ ′ are bounded (0

≤ g k ℓ ′ ≤ √ n k ℓ ′ m k ℓ ′ ) hence the sums n-1 i=0 ℓ-1 ℓ ′ =1 U i (g k ℓ ′ -U g k ℓ ′ ) = ℓ-1 ℓ ′ =1 (g k ℓ ′ -U n g k ℓ ′ ),
n ≥ 1, are uniformly bounded. If k ℓ is suciently large we thus get

1 √ n k ℓ ℓ-1 ℓ ′ =1 2n k ℓ -1 i=0 U i (g k ℓ ′ -U g k ℓ ′ ) < 1/2 ℓ . Next note that V j 1 √ n k ℓ ℓ-1 ℓ ′ =1 2n k ℓ -1 i=0 U i (g k ℓ ′ -U g k ℓ ′ ) j≥0
are martingale dierences. Hence

1 √ n k ℓ 1 √ m k ℓ 2n k ℓ -1 i=0 m k ℓ -1 j=0 U i V j ℓ-1 ℓ ′ =1 (g k ℓ ′ -U g k ℓ ′ ) 2 ≤ 1 2 ℓ . (3.3.8)
On another hand, recall that

∥g k -U g k ∥ 2 ≤ √ 2/2 k/4
. Hence choosing k ℓ suciently large we get

2n k ℓ ′ -1 i=0 U i (g k ℓ -U g k ℓ ) 2 ≤ 4 -ℓ , for all 1 ≤ ℓ ′ < ℓ.
Having constructed the sequence of k ℓ this way we thus have

1 √ n k ℓ ∞ ℓ ′ =ℓ+1 2n k ℓ -1 i=0 U i (g k ℓ ′ -U g k ℓ ′ ) 2 < 2 -ℓ . Since V j 1 √ n k ℓ ∞ ℓ ′ =ℓ+1 2n k ℓ -1 i=0 U i (g k ℓ ′ -U g k ℓ ′ ) j≥0
are martingale dierences, we get

1 √ n k ℓ 1 √ m k ℓ 2n k ℓ -1 i=0 m k ℓ -1 j=0 U i V j ∞ ℓ ′ =ℓ+1 (g k ℓ ′ -U g k ℓ ′ ) 2 ≤ 1 2 ℓ . (3.3.9)
Then, the upper bounds (3.3.8) and (3.3.9) entail that

1 √ n k ℓ 1 √ m k ℓ 2n k ℓ -1 i=0 m k ℓ -1 j=0 U i V j (I -U )(g -g k ℓ ) 2 ≤ 2 2 ℓ . (3.3.10)
Hence taking into account (3.3.7) and (3.3.10), it follows that, for f = g -U g, the sequence

(n 1 n 2 ) -1/2 n 1 -1 i=0 n 2 -1 j=0 U i V j f
cannot converge in distribution to zero. On another hand, for any p and q xed, by independence, p i=0 q j=0 E(U i V j f |F 0,0 ) = g-E(U p+1 g|F 0,0 ). But, by construction, lim p→∞ ∥U p+1 g∥ 1 = 0. Hence ∥E(U p+1 g|F 0,0 )∥ 1 is going to zero as p → ∞. Therefore condition (3.1.5) is satised.

It remains to prove that the conditions (3.1.7) and (3.1.8) are satised with

f = g -U g. With this aim, note rst that n -1/2 n-1 i=0 U i f = n -1/2 (g -U n g) → n→∞ 0 in L 1 (recall that g ∈ L 1
). Next, since the random variables (V j f ) j≥0 are independent and square integrable,

m -1/2 m-1 j=0 V j f 2 = ∥g-U g∥ 2 < ∞.
Hence both conditions in (3.1.7) are satised. On another hand, for every m xed,

1 √ n 1 √ m n-1 i=0 m-1 j=0 U i V j (g -U g) 1 ≤ 2 √ m √ n ∥g∥ 1 → n→∞ 0 ,
proving the second part of condition (3.1.8). It remains to prove its rst part. Here we use particular properties of g constructed above. We have found a sequence of n k ℓ for which there exists a positive constant c > 0 such that

(n k ℓ ) -1/2 2n k ℓ -1 i=0 U i (g - U g) 2 ≤ c 2 -ℓ + 1 . Since V j 2n k ℓ -1 i=0 U i (g -U g) j≥0 is a stationary sequence of martingale dierences in L 2 , it follows that 1 √ n k ℓ 1 √ m 2n k ℓ -1 i=0 m-1 j=0 U i V j (g -U g) 2 ≤ c 2 -ℓ + 1 ≤ 2c .
Since the upper bound is uniform for all n k ℓ , the rst part of condition (3.1.8) holds true.

3.3.3

Proof of Theorem 3.1.5

We shall use the coboundary decomposition (3.1.6). Note rst that (U i V j m) i,j is an ergodic and stationary L 2 (P) orthomartingale eld. Then, according to the CLT for ergodic elds of martingale dierences as obtained in [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF], as min(n 1 , n 2 ) → ∞, the sequence

(n 1 n 2 ) -1/2 n 1 -1 i=0 n 2 -1
j=0 U i V j m converges in distribution to a centered Gaussian random variable with variance ∥m∥ 2 2 . Theorem 3.1.5 then follows if one can prove that under the conditions of Theorem 3.1.3 and if condition (3.1.9) is satised,

then lim min(n 1 ,n 2 )→∞ ∥S n 1 ,n 2 (f ) -S n 1 ,n 2 (m)∥ 1 √ n 1 n 2 = 0 . (3.3.11)
Clearly, since

n 1 -1 i=0 n 2 -1 j=0 U i V j (I -U )(I -V )g 3 1 ≤ 4∥g 3 ∥ 1 , the convergence (3.3.11) will follow if one can prove that, as min(n 1 , n 2 ) → ∞, (n 1 n 2 ) -1/2 n 1 -1 i=0 n 2 -1 j=0 U i V j (I -U )g 1 1 + n 1 -1 i=0 n 2 -1 j=0 U i V j (I -V )g 2 1 → 0 . (3.3.12)
Since (V j (I -U )g 1 ) j≥0 and (U i (I -V )g 2 ) j≥0 are sequences of martingale dierences in L 2 (P), we shall rather prove (3.3.12) in L 2 (P) and show that

lim n→∞ ∥(I -U n )g 1 ∥ 2 √ n = 0 and lim n→∞ ∥(I -V n )g 2 ∥ 2 √ n = 0 . (3.3.13)
With this aim, we start by noticing that, for any n xed,

d 1,n := 1 √ n n-1 i=0 U i m + (I -U )g 1 is such that (V j d 1,n ) j≥0
is an ergodic and stationary sequence of L 2 (P) martingale dierences with respect to the ltration (F ∞,j ) j . Hence, by the CLT for ergodic and stationary martingales, as

N → ∞, N -1/2 N j=1 V j d 1,n → D G 1,n where G 1,n is a centered random Gaussian with standard deviation C n = ∥d 1,n ∥ 2 . Since N -1/2 N j=1 V j d 1,n N ≥1 is
uniformly integrable, by the convergence of moments theorem (see [START_REF] Billingsley | Convergence of probability measures[END_REF]Th. 3.5]) we have in particular that lim N →∞

1 √ N N j=1 V j d 1,n 1 = ∥G 1,n ∥ 1 = 2 π C n . But lim N →∞ 1 √ N N -1 j=0 V j d 1,n 1 = lim N →∞ 1 √ nN S n,N (f ) 1 . So, overall, for any n xed, lim N →∞ 1 √ nN S n,N (f ) 1 = 2 π C n ,
implying by standard arguments that there exists an increasing subsequence (n k ) tending to innity such that

lim k→∞ 1 √ n k k S n k ,k (f ) 1 - 2 π C n k = 0 . (3.3.14) Next, note that C 2 n = ∥m∥ 2 2 + 2 n E (I -U n )g 1 n-1 i=0 U i m + ∥(I-U n )g 1 ∥ 2 2 n
. But, since

(V j (I -U n )g 1 )
j≥0 is an ergodic and stationary sequence of L 2 (P) martingale dierences with respect to the ltration (F ∞,j ) j≥0 , we have, by using [1, Th. 3.4] and arguments used to get (3.

3.6), ∥(I-U n )g 1 ∥ 2 ≤ π 2 lim inf N →∞ 1 √ N n-1 i=0 N -1 j=0 U i V j (I- U )g 1 1
. Moreover, according to the coboundary decomposition (3.1.6), for any n xed,

lim inf N →∞ 1 √ N n-1 i=0 N -1 j=0 U i V j (I -U )g 1 1 = lim inf N →∞ 1 √ N n-1 i=0 N -1 j=0 U i V j (f -m) 1 .
In addition,

n-1 i=0 N -1 j=0 U i V j (f -m) 1 ≤ n-1 i=0 N -1 j=0 U i V j f 1 + √ nN ∥m∥ 2 .
So, overall, taking into account condition (3.1.9), we get

κ := lim sup n→∞ ∥(I -U n )g 1 ∥ 2 √ n ≤ π 2 ∥m∥ 2 + lim n,N →∞ 1 √ nN S n,N (f ) 1 < ∞ . (3.3.15)
Now, for any positive real A, write

C 2 n -∥m∥ 2 2 - ∥(I -U n )g 1 ∥ 2 2 n ≤ 2A √ n ∥(I -U n )g 1 ∥ 1 + 2 ∥(I -U n )g 1 ∥ 2 √ n × 1 n E n i=1 U i m 2 1 {| n i=1 U i m|>A √ n} 1/2 . (3.3.16)
Hence, using that n -1/2 ∥(I -U n )g 1 ∥ 1 → n→∞ 0 and taking into account (3.3.15) and the fact that n -1 ( n-1 i=0 U i m) 2 n≥1 is uniformly integrable, we derive that the terms in the right-hand side of (3.3.16) tend to zero by letting rst n goes to innity and after A. Therefore lim n→∞ C 2 n -∥m∥ 2 2 -

∥(I-U n )g 1 ∥ 2 2 n = 0. Assume now that κ = lim sup n→∞ 1 √ n ∥(I -U n )g 1 ∥ 2 > 0 . (3.3.17)
Then, there exists an increasing subsequence (n ′ ℓ ) ℓ≥1 tending to innity such that

lim ℓ→∞ 1 n ′ ℓ ∥(I -U n ′ ℓ )g 1 ∥ 2 = κ and then lim ℓ→∞ C 2 n ′ ℓ = ∥m∥ 2 2 + κ 2 . (3.3.18)

Extension to multidimensional index of higher dimension

To state the extension of Theorems 3.1.3 and 3.1.5 to higher dimensions, some additional notations are needed. Let d ≥ 1 and (T i ) i∈Z d be Z d actions on (Ω, A, P) generated by commuting invertible and measure-preserving transformations T εq , 1 ≤ q ≤ d. Here ε q is the vector of Z d which has 1 at the q-th place and 0 elsewhere. By U i we denote the operator in

L p (1 ≤ p ≤ ∞) dened by U i f = f • T i , i ∈ Z d . By i ≤ j, we understand i k ≤ j k for all 1 ≤ k ≤ d. Let <d>:= {1, . . . , d}. For any subset J of <d>, let U J = |J| ℓ=1 U ε ℓ
where |J| is the cardinal of J, and dene

U s J = |J| ℓ=1 U s ℓ ε ℓ for any s = (s 1 , . . . , s |J| ) in J.
We suppose that there is a completely commuting ltration (F j ) j∈Z d , i.e. there is a σ-algebra F 0 such that F i = T -i F 0 , for i ≤ j we have F i ⊂ F j and for an integrable f ,

E(E(f |F i 1 ,...,i d )|F j 1 ,...,j d ) = E(f |F i 1 ∧j 1 ,...,i d ∧j d ) . By F (k) ℓ
we denote the σ-algebra generated by all F i with i = (i 1 , . . . , i d ) with

i k ≤ ℓ and i j ∈ Z for 1 ≤ j ≤ d, j ̸ = ℓ. For σ-algebras G ⊂ F ⊂ A, by L p (F) ⊖ L p (G)
we denote the space of f ∈ L p (F) for which E(f |G) = 0 a.s.

For f a F 0 -measurable centered L 1 (P) random variable, it has been proved in Volný [START_REF] Volný | Martingale-coboundary representation for stationary random elds[END_REF]Th. 4] that the condition the series

i 1 ,...,i d ≥0 E(U i 1 ,...,i d f |F 0 ) converges in L 1 (P) (3.4.1)
ensures the existence of the following orthomartingale-coboundary decomposition:

f = m + ∅⊊J⊊< d > s∈J (I -U εs )m J + d s=1 (I -U εs )g (3.4.2)
where m, g and m J belong to L 1 (F 0 , P), L 1 ( d s=1 T εs F 0 , P) and L 1 ( s∈J T εs F 0 , P) respectively and (U i <d> m) i∈Z d and (U i J c m J ) i∈Z d-|J| are orthomartingale dierences random elds for ∅ ⊊ J ⊊<d>. Above J c =<d> \J.

For any positive integer k less than d, dene S k,d the set of all the injections from {1, . . . , k} to {1, . . . , d}. We state now the extension of Theorems 3.1.3 and 3.1.5 .

Theorem 3.4.1. Let d ≥ 1 and f a F 0 -measurable centered L 1 (P) random variable.

Let

n d = (n 1 , . . . , n d ) and S n d (f ) = n 1 -1 i 1 =0 . . . n d -1 i d =0 U i 1 ,...,i d f
. Assume that each of the transformations T εq , 1 ≤ q ≤ d, is ergodic. Suppose also that condition (3.4.1) holds and that for any integer k ∈ {1, . . . , d} and any σ in S k,d ,

lim inf n σ(1) →∞ • • • lim inf n σ(k) →∞ E n σ(1) -1 i 1 =0 . . . n σ(k) -1 i k =0 U i 1 ,...,i k f ( k i=1 n σ(i) ) 1/2 < ∞ . (3.4.3)
Then m ∈ L 2 (P) and for any set J such that ∅ ⊊ J ⊊<d>, s∈J (I -U εs )m J ∈ L 2 (P) (m and m J are dened in (3.4.2)). If in addition,

lim min(n 1 ,n 2 ,...,n d )→∞ E(|S n d (f )|) ( d i=1 n i ) 1/2 exists, (3.4.4)
then

( d i=1 n i ) -1/2 S n d (f )
converges in distribution to a centered normal variable (that can be degenerate) as min(n 1 , n 2 , . . . , n d ) → ∞.

Proof of Theorem 3.4.1. The result will follow by recurrence. Note that it holds for d = 1 and also for d = 2 as shown in the previous section. Assume that it holds for d -1 and let us prove it for d. Recall the decomposition (3.4.2) and let

m ′ = m + ∅⊊J⊆< d > 1 s∈J (I -U εs )m J , (3.4.5)
where

<d> 1 =<d> \{1} = {2, . . . , d}. Note that (U i ε 1 m ′ ) i∈Z is a stationary sequence of L 1 (P) martingale dierences w.r.t. (F i,0,...,0 ) i∈Z . Since T ε 1 is ergodic, by Theorem 3.1.2, if lim inf n→∞ n -1/2 ∥ n-1 i=0 U i ε 1 m ′ ∥ 1 < ∞, then m ′ ∈ L 2 (P)
. This follows from (3.4.2) and the fact that, by condition (3.4.3), lim inf n→∞

E(| n-1 i=0 U i ε 1 f |) √ n < ∞.
n σ(2) →∞ • • • lim inf n σ(k) →∞ E n σ(2) -1 i 2 =0 . . . n σ(k) -1 i k =0 U 0,i 2 ,...,i k m ′ ( k i=2 n σ(i) ) 1/2 < ∞ . (3.4.6)
then m ∈ L 2 (P) and, for any set J such that ∅ ⊊ J ⊊<d> 1 , s∈J (I-U εs )m J ∈ L 2 (P). By using similar arguments as those developed in the proof of Theorem 3.1.3, we infer that (3.4.6) is satised under condition (3.4.3). Hence m ∈ L 2 (P). Then, using in addition that m ′ ∈ L 2 (P), we conclude that, for any set J such that ∅ ⊊ J ⊆<d> 1 ,

s∈J (I -U εs )m J ∈ L 2 (P).
The rst part of Theorem 3.4.1 follows by using d -1 times the same arguments and replacing <d> 1 by <d> i for i = 2, . . . , d. The second part of the theorem follows by applying the CLT for ergodic orthomartingales as proved in Volný [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF] for

( d i=1 n i ) -1/2 S n d (m)
and by using similar arguments as those developed in the proof of Theorem 3.1.5. stationary and ergodic sequences of regular random elements with values in L 1 (R, µ) provided that (4.1.2) holds and

i∈Z R ∥P 0 (X i (t))∥ 2 µ(dt) < ∞, (4.1.4)
where

P 0 (•) = E (•|F 0 ) -E (•|F -1
). Her condition is in the spirit of Hannan [START_REF] Hannan | Central limit theorems for time series regression, Z. Wahrscheinlichkeitstheor und verw[END_REF]. Later, Cuny [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF]

Notations and denitions

We will use the same notations as in [START_REF] Volný | A functional CLT for elds of commuting transformations via martingale approximation[END_REF] and [START_REF] Volný | Martingale-coboundary decomposition for stationary random elds[END_REF]. We shall consider Banach-valued random elds. Hence, in all the paper, (X , |•| X ) will be a real separable Banach space. We denote by X * the topological dual of X . We shall also denote by L 0 (X ) the space of functions from Ω → X that are limits P-a.s. of simple functions. In addition, for every p ≥ 1, we dene the usual Bochner spaces L p as follows

L p (Ω, A, P, X ) = L p (X ) = Z ∈ L 0 (X ) : E (|Z| p X ) < ∞ .
For every Z in L p (X ), write ∥Z∥ p,X = (E (|Z| p X )) 1/p = ∥|Z| X ∥ p . To dene now random elds, we start by introducing Z d -actions. With this aim, we denote elements of Z d by i := (i 1 . . . , i d ). Let (T i ) i∈Z d be a Z d -action on (Ω, A, P) generated by commuting invertible and measure-preserving transformations {T e 1 , . . . , T e d }, where, for 1 ≤ i ≤ d, e i is a Z d vector with 1 at i-th place and 0 elsewhere. We denote those transformations by T (1) , . . . , T (d) . By U i we denote the operator in L p (X ) (1 ≤ p < ∞) dened by U i f = f • T i . We assume that the ltration (F k ) k∈Z d is completely commuting, i.e. there exists a σ-algebra F such that F i = T -i F, for i ≤ j we have F i ⊂ F j (i ≤ j means i q ≤ j q for all 1 ≤ q ≤ d) and for an integrable X, we have

E (E (X|F i 1 ,...,i d ) |F j 1 ,...,j d ) = E (X|F i 1 ∧j 1 ,...i d ∧j d ) ,
where i ∧ j = min {i, j} . Note that the natural ltration associated with an independent and identically distributed (i.i.d.) random eld (ξ i ) i∈Z d and dened by

F i = σ(ξ k , k ≤ i)
, is completely commuting. Completely commuting ltrations can also be constructed using stationary random elds with independent rows or columns (see [START_REF] Machkouri | A central limit theorem for stationary random elds[END_REF]).

For a xed q ∈ {1, . . . , d} and a xed ℓ ∈ Z, we denote by F (q) ℓ the σ-algebra generated by all F i with i such that i q ≤ ℓ. For every 1 ≤ q ≤ d, dene also

F (q) -∞ = i∈Z F (q) i , and F ∞ = i∈Z d F i .
For σ-algebras G ⊂ F ⊂ A and 1 ≤ p < ∞, we denote by L p (F, X )⊖L p (G, X ) the space of X ∈ L p (F, X ) for which E (X|G) = 0. We can dene projection operators

P (q) ℓ onto L p (F (q) ℓ , X ) ⊖ L p (F (q) ℓ-1 , X ) by P (q) ℓ (X) = E(X|F (q) ℓ ) -E(X|F (q)
ℓ-1 ). Those operators commute and for ℓ ̸ = k, P (q) ℓ P (q) k = 0 (see [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF]Lemma 2.4]). We now dene the projections P i = P

(1)

i 1 • • • P (d) i d onto 1≤q≤d L p (F (q) iq , X ) ⊖ L p (F (q) iq-1 , X ).
Dénition 4.2.1. Let X ∈ L 1 (X ). The stationary random elds (U i X) i∈Z d is said to be regular if X is F ∞ -measurable, and for every 1 ≤ q ≤ d, E(X|F (q) -∞ ) = 0 a.s.

Next we give the denition of ortho-martingales.

Dénition 4.2.2. Let D ∈ L 1 (X ) . We say that (U i D) i∈Z d is a eld of ortho- martingale dierences with respect to a completely commuting ltration (F i ) i∈Z d , if D is F 0 -measurable and for all i, j ∈ Z d , such that j q < i q for some q ∈ {1, . . . , d}, we have

E D • T i |F j = 0 .
In addition,

M n := n 1 i 1 =1 • • • n d i d =1 D
• T i is said to be an ortho-martingale.

Since our results are stated for random elds in Banach spaces that are 2-smooth or of cotype 2, let us recall their denitions (see [START_REF] Pisier | Martingales in Banach spaces[END_REF] for more details). We say that X is 2-smooth, if there exists L ≥ 1, such that

|x + y| 2 X + |x -y| 2 X = 2 |x| 2 X + L |y| 2 X ∀x, y ∈ X . (4.2.1)
For such a L, we say that X is (2, L)-smooth. We shall also recall the following inequality for m.d.s. in 2-smooth Banach space ([33, Prop 1]): Assume that X is

(2, L)-smooth, then for every m.d.s. (D i ) i∈Z , we have

E |D 1 + • • • + D N | 2 X ≤ 2L 2 N i=1 E |D i | 2 X for all N ∈ N . (4.2.2)
For example, for p ≥ 2 the spaces L p (R) are (2, √ p -1)-smooth. As a counterpart, we recall that a separable Banach space X is said of cotype 2 if there exists L > 0 such ance operator. We denote by G(X) a Gaussian variable having the same covariance operator as X. Dénition 4.3.3. Let G(Ω, A, P, X ) = G(X ) be the set of pregaussian random elements that are in L 2 (X ). For every X ∈ G(X ), denote ∥X∥ G(X ) = ∥X∥ 2,X + ∥G(X)∥ 2,X .

We are now in position to state the functional form of the CLT for sequences of ortho-martingale dierences with values in some Banach spaces as well as a L 2maximal inequality for the corresponding partial sums. Below and all along the paper, n → ∞ means min(n 1 , . . . , n d ) → ∞. Next theorem is the extension to the random elds setting of [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF]Prop. 3.2]. Théorème 4.3.4. Let X be a real separable Banach space that is either 2-smooth or of cotype 2. Let D ∈ L 2 (X ) be such that (U i D) i∈Z d is a eld of ortho-martingale dierences w.r.t. a completely commuting ltration (F i ) i∈Z d . Assume in addition that D ∈ G (F 0 , X ) and that at least one of the

T (i) for 1 ≤ i ≤ d is ergodic. Then, as n → ∞, {T nt (D)} t∈[0,1] d converges in distribution in C([0, 1] d , X ) (equipped
with the uniform topology) to a Brownian motion (W t ) t∈[0,1] d with covariance K D associated with D. In addition, there exists C ≥ 0, such that

max 1≤k≤n |S k (D)| X 2 ≤ C(n 1 • • • n d ) 1/2 ∥D∥ G(X ) .
(4.3.2)

Application to stationary random elds in

L p (1 ≤ p ≤ 2) Let 1 ≤ p ≤ 2.
In this section we consider a σ-nite measure space (S, S, µ) such that X := L p (S, S, µ) is separable. Recall that, since 1 ≤ p ≤ 2, L p (S, µ) is a Banach space of cotype 2. Let L q (S, µ), for q = p/(p -1), be its dual space.

In all this section, X is a random variable in L 2 (Ω, A, P, X ) and (F i ) i∈Z d with

F i = T -i F is a completely commuting ltration. Let also X i = U i X.
Corollary 4.4.3. Let p ∈ [START_REF] Billingsley | Convergence of probability measures[END_REF][START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]. Let X := (X k ) k∈Z d be a regular centered random variable in L 0 (Ω, A, P, L p (S, µ)) such that S ∥X 0 (s)∥ p 2 µ(ds) < ∞. If the following conditions hold:

S ∞ i 1 =1 • • • ∞ i d =1 1 √ i 1 • • • i d ∥E(X i (s)|F 0 )∥ 2 µ(ds) p < ∞ (4.4.6)
and

S ∞ i 1 =1 • • • ∞ i d =1 1 √ i 1 • • • i d ∥X -i (s) -E(X -i (s)|F 0 )∥ 2 p µ(ds) < ∞, (4.4
∥Y 0 ∥ 2,X < ∞ and i∈Z d |a i | < ∞.
(4.4.9)

When d = 1, our condition (4.4.8) and condition [START_REF] Gordin | The central limit theorem for stationary processes[END_REF] in [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF] have a dierent range of applications. For instance (4.4.9) is not enough for [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF]Condition (3)] to hold.

by: for every f, g ∈ L q (R, λ) (where q is the conjugate of p):

K µ (f, g) = i∈Z d E R R f (s)g(t) 1 Y 0 ≤s -F (s) 1 Y i ≤t -F (t) dsdt .
In particular, as n → ∞,

(n 1 • • • n d ) p/2 R |F n (s) -F (s)| p ds ⇒ R |G(s)| p ds.
Note that by Remark 4.4.2, condition (4.5.2) implies that X ∈ G(F 0 , L p (R, λ)). Next we give a sucient condition for (4.5.3) to hold, in terms of dependence coecients. With this aim, in the spirit of [START_REF] Dedecker | New dependence coecients. examples and applications to statistics[END_REF], dene

ϕ Y (i) = sup s∈R ∥P (Y i ≤ s|F 0 ) -P (Y i ≤ s)∥ ∞ . (4.5.4) Since P (Y i ≤ s|F 0 ) -F (s) 2 2 ≤ 2F (t)(1 -F (t))ϕ Y (i) ,
we get the following proposition: Proposition 4.5.2. Condition (4.5.3) holds as soon as (4.5.2) holds and

∞ i 1 =1 • • • ∞ i d =1 1 √ i 1 • • • i d ϕ Y (i) < ∞. (4.5.5)
Remark 4.5.3. Note that the ltration dening the coecients ϕ Y (i) is commuting, so Proposition 4.5.2 is not comparable to other results for mixing random elds such as those obtained in Bolthausen [START_REF] Bolthausen | On the central limit theorem for stationary mixing random elds[END_REF] or Dedecker [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF] where the underlying ltrations are not the same.

M n = n 1 k 1 =1 • • • n d k d =1
D k satises the following maximal inequality: there exists a positive constant C only depending on (L, p, d) such that for any positive integers n 1 , . . . , n d , m and for any λ ≥ 1,

d i=1 n -1/2 i sup 1≤k≤n |S k -M k | X p ≤ C d i=1 |k i |>m k\{k i }∈Z d-1 |P k (X 0 )| X p + Cm 2d-1 λ 2 min 1≤i≤d n 1/2 i + λ -1 min 1≤i≤d n (p-2)/(2p) i ∥|X 0 | X ∥ p + ∥|D 0 | X ∥ p ) +Cm 2d-1 1 min 1≤i≤d n (p-2)/(2p) i ∥|X 0 | X 1 {|X 0 | X >λ∥|X 0 | X ∥p} ∥ p +∥|D 0 | X 1 {|D 0 | X >λ∥|D 0 | X ∥p} ∥ p . ( 4
max 1≤k≤n |S k (D)| X 2 2 ≤ 2 2d |S n (D)| X 2 2 .
Assume rst that X is (2, L)-smooth, then applying (4.2.2) to each index, we derive

|S n (D)| X 2 2 ≤ 2 d L 2d n 1 i 1 =1 • • • n d i d =1 E |D i | 2 X .
Recall that any 2-smooth Banach space is of type 2. Therefore (4.3.2) holds by stationarity and the fact that, in Banach spaces of type 2, the norms ∥•∥ G(X ) and ∥•∥ 2,X are equivalent (see [START_REF] Ledoux | Probability in Banach Spaces: isoperimetry and processes[END_REF]Prop 9.24]). Suppose now that X is of cotype 2. Since D is assumed to be pregaussian, so is S n (D). In addition, by the orthogonality of ortho-martingale increments, we get

G(S n (D)/ √ n 1 • • • n d ) = G(D).
Hence by [START_REF] Ledoux | Probability in Banach Spaces: isoperimetry and processes[END_REF]Prop 9.25], we deduce that

∥S n (D)∥ 2,X ≤ C ∥G(S n (D))∥ 2,X = C √ n 1 • • • n d ∥G(D)∥ 2,X ≤ C √ n 1 • • • n d ∥D∥ G(X ) .
This ends the proof of (4.3.2). Now we prove the functional central limit theorem by rst proving the tightness and then the convergence of nite dimensional laws. For the tightness, the idea in [35, Proof of Prop 3.2] also applies in higher dimension. For the reader's convenience, let us give the details.

Since X is separable, then σ(D) is countably generated. Thus there exists an increasing ltration (G m ) m∈N * , such that G m is nite for every m ≥ 1 and σ

(D) = m≥1 G m . We set D m := E (D|G m ). Since G m is nite, there exist A (m) 1 , • • • A (m) km ∈ G m and x (m) 1 , • • • x (m) km ∈ X such that D m = km k=1 x (m) k 1 A (m) k . Using [35, Lemma 2.3], we have ∥D m -D∥ G(X ) -→ 0 as m → ∞.

Now, setting

Dm := P 0 (D m ) .

One can see that (U i Dm ) i∈Z d is a sequence of ortho-martingale dierences with re- spect to the completely commuting ltration (F i ) i∈Z . Then using [35, Lemma 2.2] and the fact that

E(D|F -e 1 ) = • • • = E(D|F -e d ) = 0, we deduce that (U i Dm ) m≥1 converges in G(X ) to D. By [18, Theorem 1], for every m ≥ 1, (n 1 • • • n d ) -1/2 S n,t ( Dm ) t∈[0,1] d is tight in C([0, 1] d , X
). Indeed Dm takes only a nite number of values and therefore we can work on the nite dimensional vector space generated by these values. Note that Theorem 1 in [START_REF] Volný | A functional CLT for elds of commuting transformations via martingale approximation[END_REF] is stated for reversed ortho-martingale dierences, but it is also obviously true in case of ortho-martingales in the usual sense. Now, by (4.3.2), we deduce that

1 √ n 1 • • • n d max 1≤k≤n S k ( Dm ) -S k (D) X 2 ≤ C Dm -D G(X ) -→ 0 as m → ∞ .
Hence the tightness of (n

d ). Our aim is to prove that for any m ≥ 1 and any (0, 0, . . . , 0) < t (1) < . . . < t (m) ≤ (1, 1, . . . , 1),

T nt (1) , . . . , T nt (m) ⇒ W t (1) , . . . , W t (m) , (4.7.1)
where (W t ) t∈[0,1] d is a Brownian sheet with covariance operator K D . For the reader's convenience, we shall give the complete proof in case d = 2, noticing that the general case can be proved by induction. Since either T (1) or T (2) is ergodic, let us assume from now that T (2) is (recall that T (2) is equal to T 0,1 ). Using the Cramer-Wold device, it is sucient to prove that for any m ≥ 1, any (0, 0) < t (1)

< • • • < t (m) ≤ (1, 1) and any x * 1 , • • • x * m ∈ X * , m i=1 x * i T nt (i) ⇒ m i=1 x * i W t (i) . ( 4
0 = t 0 < t 1 < • • • < t m ≤ 1, 0 = s 0 < s 1 < • • • < s m ≤ 1 and (x * i,j ) 1≤i,j≤m ∈ X * V n 1 n 2 := 1 √ n 1 n 2 m i=1 m j=1 x * i,j   [n 1 t i ] k=[n 1 t i-1 ]+1 [n 2 s j ] ℓ=[n 2 s j-1 ]+1 D • T k,ℓ   ⇒ m i=1 m j=1 x * i,j W t i ,s j + W t i-1 ,s j-1 -W t i ,s j-1 -W t i-1 ,s j , (4.7.3)
as n 1 ∧n 2 → ∞. Notice that the random variable on the right-hand side is distributed according to N (0, σ 2 ) with

σ 2 = m i=1 m j=1 (t i -t i-1 ) (s j -s j-1 ) E x * i,j (D) 2 .
Clearly, it suces to prove the convergence (4.7.3) when n 1 , n 2 → ∞ along any sequence (n r , N r ) r≥1 . Hence, let us x a sequence (n r , N r ) r≥1 such that n r , N r → ∞ as r → ∞. It remains to prove that

V r = 1 √ n r N r m i=1 m j=1 x * i,j   [nrt i ] k=[nrt i-1 ]+1 [Nrs j ] ℓ=[N -rs j-1 ]+1 D • T k,ℓ   ⇒ N (0, σ 2 ) . (4.7.4)
As in [START_REF] Volný | A functional CLT for elds of commuting transformations via martingale approximation[END_REF], the proof of (4. Lemma 4.7.1. Let ∆ i = m j=1 (s j -s j-1 )E x * i,j (D) 2 and ε > 0 . If T 0,1 is ergodic, there exist integers v > 0 (large enough) and p(v) (large enough), such that for every

n ≥ p(v) 1 v v k=1   m j=1 1 √ n [ns j ] ℓ=[ns j-1 ]+1 x * i,j (D) • T k,ℓ   2 -∆ i 1 < ε . ( 4
(2) ℓ ℓ∈Z . Let d k,ℓ (i, j) = x * i,j (D) • T k,ℓ and dene D ℓ (i, j) = (d 1,ℓ (i, j), . . . , d v,ℓ (i, j)) t .
Note that (D ℓ (i, j)) ℓ≥1 is a stationary and ergodic (since T (2) is ergodic) sequence of R v -valued martingale dierences. Therefore from the functional form of the central limit theorem for R v -valued stationary and ergodic L 2 martingale dierences, we get that for any positive integer m, and any 0 ≤ s

1 < . . . < s m ≤ 1   1 √ n [ns 1 ] ℓ=1 D ℓ (i, j), . . . , 1 √ n [nsm] ℓ=[ns m-1 ]+1 D ℓ (i, j)   converges in distribution to (G 1 , . . . , G m )
, where (G u ) 1≤u≤m are independent and centered Gaussian random variables with respective covariance matrix converges in distribution to (N k ) 1≤k≤v where N k are i.i.d random variables with common distribution N 0, m j=1 (s j -s j-1 )E x * i,j (d) 2 . Now using the notations

E G ℓ G t ℓ = (s ℓ -s ℓ-1 )E D 1 (i, j)D 1 (i, j) t = (s ℓ -s ℓ-1 ) x * i,j (d)
F i,k,n = m j=1 1 √ n [ns j ] ℓ=[ns j-1 ]+1 x * i,j (d) • T k,ℓ and V i,k,n = F 2 i,k,n -∆ i , note that ∆ i = E F 2 i,k,n = E (N 2 k ).
To soothe the notation, we will drop the index i in the rest of the proof. Let ε > 0, M > 0 and dene

A = E 1 v v k=1 V k,n 1 1 v v k=1 V k,n ≤ ε , B = E 1 v v k=1 V k,n 1 ε < 1 v v k=1 V k,n ≤ M and C = E 1 v v k=1 V k,n 1 1 v v k=1 V k,n > M .
We have

E 1 v v k=1 V k,n ≤ A + B + C . Clearly A ≤ ε. Next, B ≤ M P 1 v v k=1 V k,n > ε Since (F i,k,n ) 1≤k≤v ⇒ (N k ) 1≤k≤v , we get that, for any ε > 0, P 1 v v k=1 V k,n > ε → n→∞ P 1 v v k=1 N 2 k -∆ i > ε which converges to zero as v → ∞
C ≤ 2E h M 1 v v k=1 V k,n .
Since h M is a convex function, we deduce that

C ≤ 2 1 v v k=1 E (h M (V k,n )) .
But since for each i and k, F 2 i,k,n n≥1 is a uniformly integrable family,

lim M →∞ lim sup n→∞ E (h M (V k,n )) = 0.
So overall the lemma follows by nally letting ε → 0. .

By applying Proposition 4.6.1 to X 0 (s) and X = R, it follows that there exists a positive constant C only depending on p and d such that for any positive integers n 1 , . . . , n d , m and for any real λ ≥ 1,

(n 1 • • • n d ) -1/2 S max 1≤k≤n S k (X(s)) -S k (D(s)) p 2 µ(ds) 1/p ≤ C S d i=1 |k i |>m k\{k i }∈Z d-1 ∥P k (X 0 (s))∥ 2 p dµ(s) 1/p + Cm 2d-1 λ 2 min 1≤i≤d n 1/2 i + 1 λ S ∥X 0 (s)∥ p 2 + ∥D 0 (s)∥ p 2 dµ(s) 1/p + Cm 2d-1 S X 0 (s)1 {|X 0 (s)|>λ∥X 0 (s)∥ 2 } p 2 + D 0 (s)1 {|D 0 (s)|>λ∥D 0 (s)∥ 2 } p 2 µ(ds) 1/p := I 1 + I 2 + I 3 .
Letting rst n → ∞, then λ → ∞ and after m → ∞, the R.H.S. is tending to zero. Indeed, I 1 is tending to zero as m → ∞ by using the dominated convergence theorem and by taking into account condition (4.4.2). I 2 is tending to zero by letting rst n → ∞, then λ → ∞ and by taking into account that (4.4.2) implies that S ∥D 0 (s)∥ p 2 µ(ds) < ∞ (and then, by [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF]Lemma 2.4], that D 0 (•) is pre-Gaussian). Finally, I 3 is tending to zero by letting λ → ∞ and using the dominated convergence theorem and conditions (4. Hence it suces to prove that for any f ∈ L q (S, µ), where q is the con-

lim n→∞ max 1≤k≤n S k (X) √ n 1 • • • n d - S k (D) √ n 1 • • • n d 2,X = 0. ( 4
(n 1 n 2 ) -1/2 sup 1≤k≤n 1 1≤ℓ≤n 2 S k,ℓ -M k,ℓ X p ≤ C |u|>m,v∈Z ∥|P u,v (X 0,0 )| X ∥ p + C u∈Z,|v|>m ∥|P u,v (X 0,0 )| X ∥ p + Cm 3 λ 2 1 √ n 1 ∧ n 2 + λ -3 (n 1 ∧ n 2 ) (p-2)/(2p) ∥|X 0,0 | X ∥ p + ∥|D 0,0 | X ∥ p ) +C m 3 (n 1 ∧ n 2 ) (p-2)/(2p) ∥|X 0,0 | X 1 {|X 0,0 | X >λ∥|X 0,0 | X ∥p} ∥ p +∥|D 0,0 | X 1 {|D 0,0 | X >λ∥|D 0,0 | X ∥p} ∥ p . ( 4

.7.13)

To prove (4.7.13), the idea is to extend the decomposition (29) in [START_REF] Dedecker | Invariance principles for linear processes with application to isotonic regression[END_REF] stated in the case d = 1, to higher dimension. The idea of this decomposition is then as follows. Fix a positive integer m, and dene

M (m) k,ℓ = k i=1 ℓ j=1 D i,j • T m,m . Then we have sup 1≤k≤n 1 1≤ℓ≤n 2 S k,ℓ -M k,ℓ X ≤ sup 1≤k≤n 1 1≤ℓ≤n 2 S k,ℓ -M (m) k,ℓ X + sup 1≤k≤n 1 1≤ℓ≤n 2 M (m) k,ℓ -M k,ℓ X .
To take care of the norm of the last term we use among others Doob's maximal inequality. Next, to take care of the norm of the rst term in the right-hand side, we shall use a decomposition of X 0,0 -D 0,0 • T m,m with coboundaries and mixed coboundaries plus a rest. We refer to the forthcoming decomposition (4.7.14) for a more precise statement. In this decomposition, the term Y (m) 0,0 -Z (m) 0,0 will be considered as the rest and, once summed over a rectangle, will have at the end a negligible contribution because of the assumption (4.7.12). The rst term I in (4.7.14) will be the coboundary and the others involving g (m) and h (m) will be the mixed coboundaries in the sense that they are coboundaries in one direction and martingales in the other. As usual the coboundary structure will allow to simplify the treatment of the norm of those terms since they will behave as a telescoping sum, and next a usual truncation argument can be used. The mixed coboundaries same arguments are used in addition to Doob's maximal inequality.

We now give the details. Since X 0,0 is regular, we can write X 0,0 = i,j∈Z P i,j (X 0,0 ) a.s. Let m be a xed positive integer, and dene Observe that M n 2 = n 2 j=1 D 0,j = M ′ n 2 + M ′′ n 2 and that for any nonnegative reals a, b and ε, (a + b) p 1 {a+b≥2ε} ≤ 2 p a p 1 {a≥ε} + 2 p b p 1 {b≥ε} . Therefore

n 2 j=1 D 0,j X 1 {| n 2 j=1 D 0,j | X >2A √ n 2 } p p ≤ 2 p |M ′ n 2 | X 1 {|M ′ n 2 | X >A √ n 2 } p p + 2 p |M ′′ n 2 | X 1 {|M ′′ n 2 | X >A √ n 2 } p p .
Using that X is (2, L)-smooth, [42, Th. 2.6] and stationarity, we get

|M ′′ n 2 | X 2 p ≪ n 2 ∥|D 0,0 | X 1 {|D 0,0 | X >B} ∥ 2 p .
Next, using again that X is (2, L)-smooth and [42, Th. 2.6], we derive

|M ′ n 2 | X 1 {|M ′ n 2 | X >A √ n 2 } p p ≤ 1 A p (n 2 ) p/2 E |M ′ n 2 | 2p X ≪ 1 A p (n 2 ) p/2 n 2 j=1 ∥|d ′ j | X ∥ 2 2p p ≪ (n 2 ) p/2 A p ∥|D 0,0 1 {|D 0,0 | X ≤B} | X ∥ 2p 2p ≪ (n 2 ) p/2 B p A p ∥|D 0,0 | X ∥ p p .
So, overall, for any positive reals A and B, max We deal now with the third and fourth term in the R.H.S of decomposition (4.7.15).

For any positive real A, using stationarity, we infer that max Next noticing that for any real B, P i,v (X u+i,0 ) = P i,v (X u+i,0 1 {|X u+i,0 | X ≤B} ) + P i,v (X u+i,0 1 {|X u+i,0 | X >B} )

and, proceeding as to get (4. that for µ-almost every t, all the condition in Thereom 3.1.3 are satised. Therefor, we deduce that E m(t) 2 + E ((I -U )g 1 (t)) 2 + E ((I -V )g 2 (t)) 2 < ∞ .

(5.2.2)

For µ-almost every t, (U i V j m(t)) i,j is a stationary and ergodic elds of orthomartingale dierences, so (V j m(t)) j is a stationary and ergodic sequence of martingale dierences with nite variance. By the CLT for stationary and ergodic martingales with nite variance, as N → ∞, N -1/2 N i=1 U i m(t) converges in distribution to a centered Gaussian random variable G(t) with variance ∥m(t)∥ 2 2 . Hence, by [1,

  Next, starting from (3.4.5), and taking into account the induction hypothesis, namely: Theorem 3.4.1 holds for d -1, we infer that if for any integer k in [2, d] and any injection σ from {2, . . . , k} to <d> 1 , lim inf

7 . 4 )

 74 is based on the usual central limit theorem for trian-gular arrays of martingale dierences due McLeish (see[START_REF] Hall | Martingale limit theory and its application[END_REF] Theorem 3.2] for an easy reference). The rest of the proof then follows the lines of [18, Section 3.2] by noticing that sup 1≤i,j≤m x * i,j (D) 2 < ∞ and the following modication of[START_REF] Volný | A functional CLT for elds of commuting transformations via martingale approximation[END_REF] Lemma 4].

2 2

 2 Id, and for any i, j, k, n -1 max u≤n u ℓ=1 d k,ℓ (i, j)

  4.1) and (4.4.2). Hence (4.4.3) holds. Note now that, since 1 ≤ p ≤ 2, for X = L p (S, µ), ∥X∥ 2,X = S |X(s)| p dµ(s) )| p ∥ 2/p dµ(s)

.7. 7 )

 7 Then the convergence in distribution (4.4.4) follows by using (4.7.7) together with Theorem 4.3.4. Next to prove (4.4.5), as in the proof of [18, Th 8], we use [18, Lemma 7].

PPPPP 1 v=1P 2 P 2 PD

 122 i,j (X k,ℓ ) and θ (m) u,v = θ m,m (X u,v ) . m,ℓ (X u,b ) and g k,m (X a,v ) and h m,j (X u,0 ) -2mm,m (X u,v ) = g , for any positive integer m, the following decomposition is valid:X 0,0 -D 0,0 • T m,m := I + (g ℓ)∈Z 2 \[-m+1,m-1] k,ℓ (X 0,0 ) and Z v)∈Z 2 \[1,2m-1] m,m (X u,v ) . Let R k,ℓ = S k,ℓ -M k,ℓ . We then derive that R k,ℓ =i,j -D i,j • T m,m + (θ

D.n 1 n 2 n 1 n 2 (n 1 n 2 D 2 j=1 2 p≪ m 2 n 1 X 1 X 1 ≤ n 1 m p max 1≤ℓ≤n 2 . 2

 2222211122 i,j -D i,j •T m,m • T i,j . Since X 0,0 is regular, (U k,ℓ ) k,ℓ is an orthosubmartingale w.r.t. the completely commuting ltration (F k,ℓ ) k,ℓ . By [26, Prop. 2.2.1], it follows that (max 1≤k≤n 1 U k,ℓ ) ℓ≥1 and (max 1≤ℓ≤n 2 U k,ℓ ) k≥1 are both one parameter submartingales. Therefore, for any xed pair (n 1 , n 2 ) of natural numbers, applying twice Doob's maximal inequality, we get max Next, since X is (2, L)-smooth, using twice [42, Th. 2.6] (see also[36, Th. 2.2]), we get )| X ∥ p . Since P u,v P k,ℓ (•) = 0 for (u, v) ̸ = (i, j), it follows that max |u|>m v∈Z ∥|P u,v (X 0,0 )| X ∥ p + u∈Z |v|>m ∥|P u,v (X 0,0 )| X ∥ p . (4u,v)∈Z 2 \[1,2m-1] 2 ∥|P m,m (X u,v )| X ∥ p , |u|>m v∈Z ∥|P u,v (X 0,0 )| X ∥ p + u∈Z |v|>m ∥|P u,v (X 0,0 )| X ∥ p . (4.7.17)We handle now the rst term in the R.H.S. of decomposition (4.7.15). Note rst thatk i=1 ℓ j=1 D i,j -D i,j • T m,m = i,j . (4.7.18)Since for j xed (D i,j ) j is a martingale dierences sequence, ℓ j=1 D i,j X ℓ is a submartingale. Then, by stationarity and Doob's maximal inequality, that X is (2, L)-smooth and [42, Th. 2.6] , we derive∥|D 0,j | X ∥ 2 p ≪ m 2 n 2 ∥|D 0,0 | X ∥ 2 p . ∥|D 0,0 | X ∥ 2 p .(4.7.20)To deal with the quantity coming from the second term in the R.H.S. of (4.7.18), we rst note that for any positive real A,{max 1≤ℓ≤n 2 | ℓ+m j=m+1 D i,j | X >4A √ n 2 } p .The rst term in the R.H.S. is less than 4mA √ n 2 whereas to deal with the second one we note that, by stationarity,{max 1≤ℓ≤n 2 | ℓ+m j=m+1 D i,j | X >4A √ n 2 } p pLet B be a positive real and dene d ′ j = D 0,j 1 {|D 0,j | X ≤B} -E(D 0,j 1 {|D 0,j | X ≤B} |F 0,j-1 ) , M ′ n j = D 0,j 1 {|D 0,j | X >B} -E(D 0,j 1 {|D 0,j | X >B} |F 0,j-1 ) , M ′′ n 2 = n 2 j=1 d ′′ j .

1 √ n 2 2 √ n 1 1 /p 2 √ n 1 D 2 )≤ 4 (2m - 1 ) 4 M+ (n 1 n 2 )≪ λm 4

 1221121241424 ∥|D 0,0 | X 1 {|D 0,0 | X >B} ∥ p + m n ∥|D 0,0 | X 1 {|D 0,0 | X >B} ∥ p + m n B A ∥|D 0,0 | X ∥ p . (4.7.22) Let λ > 0. Starting from (4.7.18), taking into account (4.7.19)-(4.7.22) and selectingA = λ 2 ∥|D 0,0 | X ∥ p and B = A/λ, we derive max i,j -D i,j • T m,m ∥|D 0,0 | X 1 {|D 0,0 | X >λ∥|D 0,0 | X ∥p} ∥ p 0 | X ∥ p . (4.7.23)Next note that, for any positive real M ,∥|P i,j (X k,ℓ 1 {|X k,ℓ | X >M } )| X ∥ p .Hence, by stationarity, setting M = λ∥|X 0,0 | X ∥ p , we getmax ∥|X 0,0 | X ∥ p + m 4 (n 1 n 2 ) 1/p ∥|X 0,0 | X 1 {|X 0,0 | X >λ∥|X 0,0 | X ∥p} ∥ p .(4.7.24)

PX 1 1 n 1 i=1P

 111 m+i,v (X u+i,b )•T 0,ℓ+1 X p ≤ 4(2m -1) 3 A √ n 1 + (2m -1)(n 2 + 1) {| k i=1 P i,v (X u+i,0 )| X >4A √ n 1 } p .Note that k i=1 P i,v (X u+i,0 ) X k≥1 is a submartingale. Therefore, by [15, Corollary 2.10] and stationarity,A(n 1 , u, v) ≤ 2p p -i,v (X u+i,0 ) X 1 {| n 1 i=1 P i,v (X u+i,0 )| X >2A√ n 1 } p .

7 . 21 )≪ m 3 λ 2 √ n 1

 72121 and selecting A = λ 2 ∥|X 0,0 | X ∥ p and B = A/λ, we infer that, for any λ > 0∥|X 0,0| X ∥ p + m 3 (n 1 ) 1/2 (n 2 ) 1/p ∥|X 0,0 | X 1 {|X 0,0 | X >λ∥|X 0,0 | X ∥p} ∥ p + m 3 (n 1 ) 1/2 (n 2 ) 1/p λ ∥|X 0,0 | X ∥ p . k+1,j X p ≪ m 3 λ 2 √ n 2 ∥|X 0,0 | X ∥ p + m 3 (n 1 )

( 4 . 7 .=∥E ( 1 X 1 First

 4711 [START_REF] Maxwell | Central limit theorems for additive functionals of Markov chains[END_REF] holds:X 0 -D 0 • T m = ∅⊊J⊆⟨d⟩ s∈J c (I -U s )g k∈Z d \[-m+1,m-1] d P k (X 0 ) and Z k ≤t |F 0,0 ) -F (t)∥ 1As a direct consequence we get the following corollary Corollary 5.1.3. Let B(t) = F (t)(1 -F (t)). Assume that k≥1 min(α k , B(t))µ(dt) < ∞ .(5.1.15)Then the conclusion of theorem 5.1.1 holds. , we will prove that m ∈ L 2 (X ). Now notice that by (5.1.1) the following decomposition holds for µ-almost every t f (t) = m(t) + (I -U )g 1 + (I -V )g 2 + (I -U )(I -V )g 3 (5.2.1)

  k,ℓ∈Z est un champ aléatoire de diérences d'ortho-martingale pour la ltration complètement commutante (F i,j ) i,j∈Z . Si T ou S est ergodique alors (n 1 n 2 ) -1/2 S n 1 ,n 2 (X) converge en loi vers une loi normale centrée de variance ∥X∥2 2 .

  dans le cadre des martingales. Puis à l'aide d'une approximation ortho-martingale, nous en déduirons le WIP pour des champs strictement stationnaires et ergodiques à valeurs dans L p (S, µ), 1 ≤ p ≤ 2, sous une condition projective de type Hannan.

Les trois chapitres suivants de cette thèse détaillent les résultats énoncés dans l'introduction et font l'objet des trois articles suivants : 1. H. M. Lin, On the weak invariance principle for non-adapted stationary random elds under projective criteria, Stoch Dyn, 22 -05 -2250013, 2022. https://doi.org/10.1142/S0219493722500137. 2. H. M. Lin, F. Merlevède, D. Volný, On the central limit theorem for stationary random elds under L 1 -projective condition. Electron. Commun. Probab. 271 -12, 2022. https://doi.org/10.1214/22-ECP486.

, and (b) holds, which implies that σ n 1 ,n 2 / √ n 1 n 2 is a 2p-svf (by us- ing Proposition

  2.2.4) and the same is true fors n 1 ,n 2 / √ n 1 n 2 .

	ditions hold: as n 1
	Below we give sucient conditions for C 1 (s n 1 ,n 2 )(c) to hold.
	Proposition 2.2.6. Condition C 1 (s n 1 ,n 2 )(c) is satised as soon as the following con-

  Suppose that s [n 1 t 1 ],[n 2 t 2 ] /s n 1 ,n 2 is bounded for any

	). Hence, Theorem 2.2.2 together with Proposition 2.3.1 give the
	following weak invariance principle.
	Théorème 2.3.5.

  holds then, by Remark 2.2.3, we derive that s -1 n 1 ,n 2 S n 1 ,n 2 converges in distribution to E (m 2 )N , provided that one of the transformations T (1) or T (2) is ergodic.Assume that the conditions C 1 (σ n 1 ,n 2 )(a) and C 1 (σ n 1 ,n 2 )(b) hold. To prove σ n 1 ,n 2 is a 2p-svf, it suces to prove (2.2.1). Note rst that the following decomposition holds: for every non-negative integers k and ℓ,

	2.6.2	Proof of Proposition 2.2.4

  The rst part of the proof is quite direct. Indeed, let us denote the double sums in the

	2.6.3	Proof of Proposition 2.2.6
	norm of the conditions (2.2.5) and (2.2.6) by a, b, c, d, and the quantities appearing
	in the left hand side of the conditions (2.2.5) and(2.2.6) by A, B, C, and D. For
	example:
		.6.8)
	We infer that if (2.6.7) and (2.6.8) hold then (2.2.1) can be extended to (2.2.3). Next,
	it is easy to see that (2.6.7) and (2.6.8) hold as soon as σ [x],[y] → ∞ as [x] ∧ [y] → ∞,
	and (2.2.2) is satised.

  The inequality (2.6.21) comes from inequality (3.6) in[START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF] with X i = n 2 j=1 d i,j by taking the expectation and by noticing that (X i ) i≥1 is a sequence of martingale dierences. Next (2.6.22) comes from (2.6.21) by taking account that x 2 1 |x|>λ ≤

	2.7.2	Proof of Lemma 2.6.3
	4 |x| -	λ 2	2 +

  3.1.6) are in L 2 (P). , n 2 ) → ∞, the limiting distribution behavior of (n1 n 2 ) -1/2 S n 1 ,n 2 (f )is the same as that of the orthomartingale part (n 1 n 2 ) -1/2 S n 1 ,n 2 (m). In other terms one can wonder if assuming the conditions of Theorem 3.1.3 is enough to ensure that the coboundaries' behavior, i.e. (n1 n 2 ) -1/2 (S n 1 ,n 2 (f ) -S n 1 ,n 2 (m)) is negligible for the convergence in distribution. Surprisingly the answer to this question is negative as shown by the next counterexample. Theorem 3.1.4. There exist a probability space (Ω, A, µ), a function g ∈ L 1 (µ), measurable with respect to a σ-algebra F 0,0 ⊂ A and bijective bimeasurable ergodic transformations T and S such that f = (I -U )g is in L 2 (µ), satises the conditions (3.1.5), (3.1.7) and(3.1.8) but such that (n 1 n 2 ) -1/2 S n 1 ,n 2 (f ) does not converge in distribution to zero as min(n 1 , n 2 ) → ∞.

	Compared to the case of random sequences a natural question is then to wonder
	if condition (3.1.5) together with conditions (3.1.7) and (3.1.8) are sucient to ensure
	that, when min(n 1

This result proves a drastically dierent behavior for the case of random elds with dimension d ≥ 2 compared to the case of random sequences (d = 1) for which the coboundary is negligible for the convergence in distribution as soon as (3.1.1) is assumed. Let us also indicate that even if (3.1.1) is reinforced to a convergence in L p (µ) for some p ∈ [1, 2) this is still not enough for (n 1 n 2 ) -1/2 S n 1 ,n 2 (f ) and

  n 1 ,n 2 (f ) converges in distribution to a centered normal variable (that can be degenerate).

	lent to
	lim sup	lim sup
	n→∞	k→∞
	Remark 3.1.6. Under the conditions of Theorem 3.1.3, condition (3.1.9) is equiva-

  has proved the CLT and its functional form for stationary and ergodic sequences of m.d.s. with values in more general Banach spaces than L 1 , namely cotype 2 or 2-smooth Banach spaces. Next, again with the help of a martingale approximation, he derived sucient conditions in the spirit of Maxwell-Woodroofe[START_REF] Maxwell | Central limit theorems for additive functionals of Markov chains[END_REF] ensuring the weak invariance principle for a stationary and ergodic sequence with values in either 2-smooth or cotype 2 Banach spaces. Note that when applied to the L 1 -space, his condition and (4.1.4) have a dierent range of applications.The rst aim of this paper is to generalize the functional CLT for martingale dierences sequences with values in some Banach spaces as stated in[START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF] Prop 3.2] to higher dimension in the sense of multi-indexed sequences. Recall that there is no natural ordering in higher dimension and, in this paper, we shall use the notion of

	to Section 4.7.
	completely commuting ltration (see Section 4.2 for a denition of this notion). It fol-
	lows that the generalization of [35, Prop 3.2] will mean extending the functional CLT
	to ortho-martingale sequences with values in a 2-smooth or cotype 2 Banach spaces
	(see Section 4.3). Then, in Section 4.4, with the help of a suitable ortho-martingale
	approximation, we derive a functional CLT for L p -valued stationary random elds,
	where p ∈ [1, 2]. As in [9], our conditions are in the spirit of Hannan [6]. Hence, our
	theorem 4.4.1 can be viewed as a generalization of the results stated in [9] in several
	directions. First, it proves the functional form of [, Th 2.3]. Secondly, it extends it
	to multi-indexed random sequences. Finally, it considers the case of variables with
	values in a broader class of Banach spaces. As an application, in Section 4.5, we give
	asymptotic results for the L p -distances, 1 ≤ p ≤ 2, between the common distribution
	function and the corresponding empirical distribution function for stationary random
	elds. Section 4.6 is devoted to a suitable maximal ortho-martingale approximation
	in 2-smooth Banach spaces and has interest in itself. All the proofs are postponed

  Condition(4.4.6) together with(4.4.7) imply that X is regular. Comment 4.4.5. Using Proposition 4.6.1, a similar result as Theorem 4.4.1 can be obtained when X ∈ L 0 (Ω, A, P, X ) where X = L p (S, µ) with p > 2. More precisely if ∥X 0 ∥ 2,X < ∞ and Note that if X 0 (s) = i∈Z d a i Y -i (s) where (Y i ) i∈Z d is an ortho-martingale dierences,(4.4.8) holds as soon as

									.7)
	then (4.4.2) is satised.
	Remark 4.4.4. i∈Z d		S	|P 0 (X i (s))| p µ(ds)	1/p	2	< ∞,	(4.4.8)
	then	d
	lim n→∞	i=1	n	-1/2 i	sup 1≤k≤n	|S k -M k | X	2	= 0 ,
	and the convergence (4.4.4) holds.

  Comment 4.5.4. Note that by Item 3 in [38,Lemma 2], we get that if the distribution function F of Y 0 is continuous then, for any i ≥ 0, Proposition 4.6.1. Let X be a (2, L)-smooth Banach space. Let p ≥ 2 and X 0 be a regular r.v. with values in X such that ∥|X 0 | X ∥ p < ∞. Assume that Then setting D k = i∈Z d P k (X i ), (D k ) k∈Z d are stationary L p (X ) ortho-martingale dierences with respect to (F k ) k∈Z d and the corresponding ortho-martingale

	∥|P 0 (X i )| X ∥ p < ∞ .	(4.6.1)
	i∈Z d

  Similarly to Remark 4.4.2, note that when X is regular and in L 1 (Ω, A, P, X ) then (4.6.1) implies that ∥|X 0 | X ∥ p < ∞. Remark 4.6.3. A careful analysis of the proof of Proposition 4.6.1 reveals that when X is of cotype 2, then a similar maximal inequality to that stated in (4.6.2) can be derived but only for p = 2. Indeed it suces to replace in the proof the martingale inequality (4.2.2) by the inequality (4.3.2) valid for martingales with values in Banach spaces of cotype 2. Hence, when X 0 is a regular r.v. with values in G(X ) where X is of cotype 2 and if i∈Z d ∥P 0 (X i )∥ G(X ) < ∞ then, when p = 2, Inequality (4.6.2) remains valid provided we replace in the right-hand side the norm ∥| • | X ∥ p by the pre-Gaussian norm ∥ • ∥ G(X ) .

	.6.2) The idea of this proof is essentially the same as the proof of [35, Prop 3.2], except The proof of this result is postponed to Section 4.7.4. 4.7.1 Proof of Theorem 4.3.4 for the convergence of nite dimensional laws. We rst prove (4.3.2). By the Cairoli's strong (p, p) inequality for ortho-submartingales Remark 4.6.2. 4.7 Proofs [26, Th. 2.3.1], we have

  .7.2)With this aim we shall use similar arguments as those developed in [18, Section 3.2] and, in a sake of clarity, we shall give most of the details. Notice rst that m i=1 x * i T nt (i) can be written as a weighted sum over disjoint and adjacent rectangles. Hence proving (4.7.2) is equivalent to show that for any positive integer m, any

  .7.5) Proof of Lemma 4.7.1. Note rst that for any xed i, j ∈ N and k ∈ Z, we have that x * i,j (D) • T k,ℓ ℓ∈Z is a sequence of martingale dierences with respect to the ltration F

  by the law of large numbers. Let us now deal with C. Letting h M (x) = |x| -M 2 + , where x + = x1 x>0 , and noticing that |x| 1 |x|>M ≤ 2h M (x), we get

  1/p (n 2 ) 1/2 ∥|X 0,0 | X 1 {|X 0,0 | X >λ∥|X 0,0 | X ∥p} ∥ p + m 3 (n 1 ) 1/p (n 2 ) 1/2 λ ∥|X 0,0 | X ∥ p . (4.7.26) Starting from decomposition (4.7.15) and considering the upper bounds (4.7.16), (4.7.17), (4.7.23), (4.7.24), (4.7.25) and (4.7.26), the inequality (4.7.12) follows. Let us now indicate the main argument to extend (4.7.13) to dimension d > 2 and then get (4.6.2). By simple induction we infer that the following extension of

(s n ).One can see this as an non-adapted case of Theorem 7 in[START_REF] Peligrad | Martingale approximations for random elds[END_REF]

ϕ Y (i) ≤ K F (Y i ) -F (Y * i ) ∞ ,(4.5.6)

Remerciements

La rédaction des remerciements marque la n des trois années consacrées à la thèse, qui a été une expérience exceptionnelle et unique. Une occasion de se remémorer des bons souvenirs et un moment personnel pour exprimer mes reconnaissances.

This follows from the invariance principle for elds of stationary ortho-martingale dierences (see Theorem 1 in [START_REF] Volný | A functional CLT for elds of commuting transformations via martingale approximation[END_REF]), provided that for any (t 1 , t 2 ) ∈ [0, 1] 2 , lim

(2.6.34)

Note that Theorem 1 in [START_REF] Volný | A functional CLT for elds of commuting transformations via martingale approximation[END_REF] is stated for reversed ortho-martingale dierences, but it is also obviously true in case of ortho-martingales in the usual sense. Now in order to prove ( We shall apply Theorem 2.3.5, and then we only need to show that conditions C 1 (s n 1 ,n 2 )(c) and C 2 (s n 1 ,n 2 ) hold. We rst prove C 2 (s n 1 ,n 2 )(a), and C 2 (s n 1 ,n 2 )(b). Using Remark 2.3.3, they hold as soon as C 3 (s n 1 ,n 2 ) is satised. Note that P 0,0 (X i,j ) = a i,j ξ 0,0 . Therefore C 3 (s n 1 ,n 2 ) is implied by (2.4.2), (2.4.3), and (2.4.4).

We prove now that C 2 (s n 1 ,n 2 )(c) holds with u i,j = |a i,j |. SinceP 0,0 (X i,j ) = a i,j ξ 0,0 and s n 1 ,n

But, using once again the coboundary decomposition (3.1.6), note that

Birkho's theorem in L 1 (P) implies that lim n→∞

Using in addition that g 3 is in L 1 (P), we get

Chapter 4

On the weak invariance principle for ortho-martingale in Banach spaces.

Application to stationary random elds

In this chapter, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random elds in L p , 1 ≤ p ≤ 2, under a condition in the spirit of Hannan.

As an application, we get an asymptotic result for the L p -distances (1 ≤ p ≤ 2) between the common distribution function and the corresponding empirical distribution function of stationary random elds.

Introduction

Let (X i ) i∈Z be a stationary random sequence. If one can represent (X i ) i∈Z in the form

where (d i ) i∈Z is a stationary sequence of martingale dierences, and (ζ i ) ∈Z is a coboundary, which means that it can be written as ζ i = θ i -θ i-1 with (θ i ) i∈Z a stationary random sequence, then one may derive the CLT as well as other limit theorems from the corresponding ones for the martingale dierences (d i ) i∈Z . This method is usually called the martingale approximation, but is also known as Gordin's method (see [START_REF] Gordin | The central limit theorem for stationary processes[END_REF]). Note that, as a variation of this method, it is not necessary that the sequence (ζ i ) ∈Z is a coboundary, but only is negligible, for instance in probability, under a suitable normalisation. This kind of approach has been developed in many papers. We refer to Chapter 4 in Merlevède et al. [START_REF] Merlevède | Functional Gaussian Approximation for Dependent Structures[END_REF] for a survey concerning Gaussian approximation via martingale methods.

In this paper, we will use an adaptation of the martingale approximation method to prove the CLT and its functional form for the partial sums associated with multiindexed sequences (also called random elds) with values in some Banach spaces X .

Let us rst recall some recent results concerning the limiting behavior of the partial sums associated with random sequences in Banach spaces.

Let (Ω, A, P) be a probability space, T be an invertible bi-measurable measure preserving transformation on Ω. Let F ⊂ A be a σ-algebra such that T -1 F ⊂ F, and dene a non-decreasing ltration F i = T -i (F). We also dene the stationary sequence of random elements X := (X i ) i∈Z by X i = X 0 • T i . Below, let us consider the Banach space X = L 1 (R, µ), where µ is a σ-nite measure on the real line, and L ∞ (R, µ) be its dual space. When (X i ) i∈Z is a L 1 -valued stationary ergodic sequence of martingale dierences (m.d.s.), Dede [START_REF] Dede | An empirical central limit theorem in L1 for stationary sequences[END_REF] proved the following CLT: If

Here and along the paper, ⇒ stands for the convergence in distribution and G is a centered Gaussian random variable with covariance operator K X (see denition 4.3.1). Then, with the help of a martingale approximation, she extended the CLT to that for every independent random variables D

2) holds in the reverse direction. As an example, note that for

Functional CLT for ortho-martingales in Banach spaces

Let X be a random element from Ω to X . Dene its associated partial sum S n (X) by

In addition, for t

Before stating the main result of this section, as in [START_REF] Cuny | Invariance principles under the MaxwellWoodroofe condition in Banach spaces[END_REF], we need to recall the denitions of Gaussian and pregaussian random elements.

Dénition 4.3.1. Let K be a bounded symmetric bilinear operator from X * × X * to R. We say that K = K X is the covariance operator associated with X if

Dénition 4.3.2. The random variable W ∈ L 0 (X ) is said to be Gaussian if, for every x * ∈ X * , x * (W ) has a normal distribution. We say that a random variable X ∈ L 0 (X ) such that for every x * ∈ X * , E (x * (X) 2 ) < ∞ and E (x * (X)) = 0, is pregaussian, if there exists a Gaussian variable W ∈ L 0 (X ) with the same covari- Assume also that

Then there exists a stationary random eld of ortho-martingale dierences

Suppose, in addition, that at least one of the transformations

where W is a X -valued centered Brownian motion with covariance operator K D that can be dened as follows: for all f and g in L q (S, µ) (where q is the conjugate of p),

where

Remark 4.4.2. Note that by [35, Lemma 2.4], condition (4.4.1) implies that X ∈ G(L 1 (S, µ)). Note also that when X is regular and in L 1 (Ω, A, P, L 1 (S, µ)) then the (reverse-)martingale convergence theorem in the Banach-valued case (see e.g. [START_REF] Pisier | Martingales in Banach spaces[END_REF]) ensures that (4.4.2) implies (4.4.1).

Next result gives sucient conditions for (4.4.2) to hold.

Application to empirical processes

Let Y be a real random variable which is F 0 -measurable. For every i ∈ Z d , we dene

We also denote by F n the empirical distribution function of (Y i ) i∈Z d :

For 1 ≤ p ≤ 2, we are interested in deriving the asymptotic behavior of and

Suppose, in addition, that at least one of the transformations

where Y * i is a random element distributed as Y i and independent of F 0 . To give an example, let us consider d = 2. Let (ε i,j ) (i,j)∈Z 2 be a iid random eld such that P(ε 0,0 = 0) = P(ε 0,0 = 1) = 1/2, and dene F i,j = σ(ε u,v , u ≤ i, v ≤ j). Let f (0, 0) = 0 and f (i, j) = (i + j)(i + j + 1)/2 + i for any (i, j) ∈ N 2 \{0, 0}. Then f is a one-to-one onto map from N 2 to N. Let now U k,ℓ = i≥0,j≥0 2 -f (i,j)-1 ε k-i,ℓ-j . Clearly each of the U k,ℓ 's is distributed as i≥0,j≥0 2 -f (i,j)-1 ε -f (i,j),0 and then as

, and considering (ε * i,j ) (i,j)∈Z 2 an independent copy of (ε i,j ) (i,j)∈Z 2 , we dene

Clearly U * k,ℓ is distributed as U k,ℓ and is independent of F 0,0 . Moreover

Then, according to Proposition 4.5.2, Condition (4.5.3) holds provided (4.5.2) is satised which requires in our case that a < p/2.

Ortho-martingale approximation in Banach spaces

The following result allows to derive a useful ortho-martingale approximation in 2smooth Banach spaces. It is an extension to multidimensional index sets and to 2-smooth Banach spaces of the estimate (3.3) in [START_REF] Dedecker | Rates in the strong invariance principle for ergodic automorphisms of the torus[END_REF]. It can also be viewed as an extension in several directions of [45, Th.

.

From Lemma 4.7.2 with u i,j = ∥P -i,-j (X 0,0 (s))∥ 2 , we get:

But by orthogonality and regularity

which is nite by condition (4.4.6) in case d = 2. It remains to prove that:

and

. (4.7.10)

But, for any j ≥ 0, by stationarity,

Next, by orthogonality,

.

But, note that,

Hence, by orthogonality, it follows that In the proof the notation a ≪ b means that there exists a universal constant C (here only depending on (L, p, d)) such that a ≤ Cb.

For the reader's convenience, let us consider the case d = 2 (the case d = 1 is even simpler and for d > 2 some indications are given at the end of the proof). When d = 2, (4.6.1) reads as i,j∈Z ∥|P 0,0 (X i,j )| X ∥ p < ∞ .

(4.7.12)

Then D k,ℓ = i,j∈Z P k,ℓ (X i,j ). Moreover (D k,ℓ ) k,ℓ is a stationary eld of L p (X ) ortho-martingale dierences with respect to the ltration (F k,ℓ ) (k,ℓ)∈Z 2 . In addition, the corresponding ortho-martingale is

j=1 D i,j . Next (4.6.2) reads as: there exists a positive constant C only depending on (L, p) such that for any positive integers n 1 , n 2 , m and for any real λ ≥ 1, g m J let us describe them for some given sets J: for J = J 1 = {1} we have g (m)

and so on. Inequality (4.6.2) then follows using the decomposition (4.7.27) and the arguments used to prove (4.7.13).

Chapter 5

Appendix

In this appendix, we give some related results between chapter 2 and 3. Namely the TCL under L 1 condition in L 1 (S, S, µ).

TCL under L 1 condition in Banach space

We rst recall some notations. Let (S, S, µ) be a σ-nite measure space such that

Let L ∞ (S, µ) be its dual space. Let T and S be commuting invertible and measure-preserving transformations. By U (resp V ) we denote the operator in L p (X ) dened by U f = f • U (resp V f = f • S) for any random variable f . Now for a stationary random variable f ∈ L 1 (X ) if the following condition holds for every µ-almost every t

Then the coboundary decomposition holds,

where m, g 1 , g 2 and g 3 are in L 1 (X ), (U i V j m(t)) is a stationary eld of orthomartingale dierences, (V j g 1 (t)) j is a stationary martingale dierences sequence with respect to the ltration (F ∞,j ) j , and (U i g 2 (t)) i is a stationary martingale dierences sequence with respect to the ltration (F i,∞ ) i . Now setting

and

We are now in position to give the following result.

Théorème 5.1.1. Let f be a measurable centered L 1 (X ) random variable. Let

. Assume that for µ-almost every t condition (5.1.1) holds C i (t) < ∞ and C i (t)µ(dt) < ∞ for every i ∈ {1, 2, 3, 4} .

(5.1.7)

In addition, if

√ n 1 n 2 µ(dt) exists .

(5.1.8)

Then, the random variables m, (I -U )g 1 and (I -V )g 2 dened in (5.1.2) are in L 2 (X ), and as min(n 1 , n 2 ) → ∞, (n 1 n 2 ) -1/2 S n 1 ,n 2 (f ) converges in distribution to a centered normal variable.

As an immediate consequence of the Theorem 5.1.1, we get the following Corollary.

Corollary 5.1.2. Assume that (5.1.8) and (5.1.1) hold and

(5.1.9)

Then the conclusion of Theorem 5.1.1 holds.

Empirical distribution

In this section, S = R and µ is a σ-nite measure on R equipped with the Borel σ-eld. Let X = (X k ) k∈Z be a stationary and ergodic sequence of real-valued random variables. Let X ℓ , ℓ ∈ Z be independent copies of X.

As an example, let us consider (ξ k,ℓ ) k,ℓ∈Z be an i.i.d random eld, let h be a measurable function and (X 0,ℓ ) ℓ∈Z a sequence of i.i.d. random variables such that (X 0,ℓ ) ℓ∈Z is independent of (ξ k,ℓ ) k≥1 . Then for any ℓ ∈ Z and any k ≥ 1 we dene

For each ℓ xed, (X k,ℓ ) k≥1 forms a Markov chain and assume that it has a stationary distribution. Next, from the stationary random eld (X k,ℓ ) we dene Y k,ℓ (t) = 1 X k,ℓ ≤t -F (t) ∀t ∈ R, where F is the cumulative distribution function of X 0,0 . Let F i,j = σ {X k,ℓ , k ≤ i, j ≤ ℓ}. In this setting, for non negative integers k and ℓ,

otherwise .

(5.1.10)

Then conditions (5.1.1) and (5.1.9) hold as soon as

Hence by the convergence of moments Theorem,

2 is uniformly integrable. By the convergence of moments Theorem again, it follows that for µalmost every t,

.1.14)

To end the proof of (5.1.8), note that by stationarity,

which is nite by (5.1.12).

Th. 3.4] and noticing that E|G(t)| = ∥m(t)∥ 2 2/π, we get

(5.2.4)

Following the prof of Theorem 3.1.5, we deduce that lim inf