Lucas Reding
email: lucas.reding@emse.fr

Na Zhang
email: nzhang@towson.edu

On the Quenched Functional Central Limit Theorem for Stationary Random Fields under Projective Criteria

Keywords: random fields, quenched functional central limit theorem, ortho-martingale approximation, projective condition. Mathematics Subject Classification (2020): 60G60, 60F05, 60F17, 60G42, 60G48, 41A30

, the authors have already proven a quenched functional CLT however the assumptions were not optimal (as they require the existence of a 2 + δ-moment). In this article, we establish the results under weaker assumptions, namely we only require an Orlicz space condition to hold. A Rosenthal type inequality for this Orlicz space is also derived and used to obtain a sufficient condition analogous to that of Theorem 4.4 in the work of Zhang et al. [36].

Introduction

Developments within the Markovian theory led to the question of the conditions under which a central limit theorem could be derived for Markov chains; in particular what restrictions were sufficient on the initial distribution and the transition operator to have this kind of result. Seminal results were obtained by Gordin and Lifšic [START_REF] Gordin | The central limit theorem for stationary Markov processes[END_REF] (see also Borodin and Ibragimov [2] or Derriennic and Lin [START_REF] Derriennic | The central limit theorem for Markov chains with normal transition operators started at a point[END_REF]) for Markov chains endowed with the stationary measure as their initial distribution as well as Kipnis and Varadhan [START_REF] Kipnis | Central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions[END_REF] (see also Derriennic and Lin [START_REF] Derriennic | The central limit theorem for Markov chains with normal transition operators started at a point[END_REF]) for additive functionals of reversible Markov chains. Additionally, Derriennic and Lin [START_REF] Derriennic | The central limit theorem for Markov chains with normal transition operators started at a point[END_REF] also obtained a CLT for Markov chains starting from a fixed point (in other words, endowed with δ x , the Dirac measure at the state x, as their initial distribution). This kind of theorems are called quenched CLT. Another way of representing these results is to consider a fixed past and to study the convergence in distribution with respect to that past. The difficulties during the proof arise from the fact that this fixed past causes the process to not be stationary anymore. An extensive literature exists on the subject, one can cite the following works by Barrera et al. [START_REF] Barrera | On the functional CLT for stationary Markov chains started at a point[END_REF], Cuny and Peligrad [START_REF] Cuny | Central limit theorem started at a point for stationary processes and additive functional of reversible[END_REF], Cuny and Merlevède [START_REF] Cuny | On martingale approximations and the quenched weak invariance principle[END_REF], Cuny and Volný [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF], Dedecker et al. [START_REF] Dedecker | A quenched weak invariance principle[END_REF], Peligrad [START_REF] Peligrad | Quenched Invariance Principles via Martingale Approximation; in Asymptotic laws and methods in stochastics[END_REF] and Volný and Woodroofe [START_REF] Volný | Quenched central limit theorems for sums of stationary processes[END_REF]. Note that some counter examples to quenched central limit theorems under specific conditions were found by Ouchti and Volný [START_REF] Ouchti | A conditional CLT which fails for ergodic components[END_REF] and Volný and Woodroofe [START_REF] Volný | An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process[END_REF]. Functional versions of these quenched central limit theorems, also called quenched weak invariance principles, have also been the subject of numerous research articles such as the ones by Barrera et al. [START_REF] Barrera | On the functional CLT for stationary Markov chains started at a point[END_REF], Cuny and Merlevède [START_REF] Cuny | On martingale approximations and the quenched weak invariance principle[END_REF], Cuny and Volný [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF] and Peligrad [START_REF] Peligrad | Quenched Invariance Principles via Martingale Approximation; in Asymptotic laws and methods in stochastics[END_REF].

Random fields naturally appear as a generalization of sequences of random variables, however extending the one-dimensional results to greater dimension is much harder than one would think. The first problem we are faced with is to correctly define the notion of past trajectory. The approach we have implemented in this paper is to use the notion of commuting filtrations. In particular, this property is satisfied by filtrations generated by fields of independent random variables or even by fields with independent columns (or equivalently independent rows). As a lot of process can be expressed as a functional of i.i.d. random variables, these type of filtrations are quite common and merit interest. A lot of work has been done under commuting filtrations (see Volný [START_REF] Volný | A central limit theorem for fields of martingale differences[END_REF] and Cuny et al. [START_REF] Cuny | A functional central limit theorem for fields of commuting transformations via martingale approximation[END_REF]).

As usual, we will require some kind of dependency condition on the studied field. Namely, in this paper, we will use Hannan's projective condition [START_REF] Hannan | Central limit theorems for time series regression[END_REF]. The problem we are interested in has been studied by Cuny and Volný [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF] for time series but it has yet to be investigated for higher dimensions, which is the purpose of this article. Though the problem we focus on hasn't been studied yet, one can note that fields satisfying Hannan's condition have been quite extensively studied and numerous CLTs and functional CLTs (both annealed and quenched) have been obtained. One could refer to the following: Volný and Wang [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF], Klicnarová et al. [START_REF] Klicnarová | Limit theorems for weighted Bernoulli random fields under Hannan's condition[END_REF] and Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF].

The proof of the main theorem in this paper is based upon the use of a martingale-coboundary decomposition that can be found in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] (some more recent and general results can be found in [START_REF] El Machkouri | Orthomartingale-coboundary decomposition for stationary random fields[END_REF][START_REF] Volný | Martingale-coboundary decomposition for stationary random fields[END_REF][START_REF] Giraudo | Invariance principle via ortho-martingale approximation[END_REF], see also [START_REF] Gordin | Martingale-coboundary representation for a class of stationary random fields[END_REF]) as well as the central limit theorem and the weak invariance principle established by Peligrad and Volný [START_REF] Peligrad | Quenched invariance principles for ortho-martingale-like sequences[END_REF] for ortho-martingales. Once the main theorems are established, we derive corollaries in the spirit of the results obtained by Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF]. As shown by the previous results in the literature, it will be required to address two situations separately: first when the summations are done over cubic regions of Z d and, after that, when the regions are only required to be rectangular.

In the previous work of Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF], the Rosenthal inequality for Lebesgue spaces (see Hall and Heyde [START_REF] Hall | Martingale Limit Theory and Its Application[END_REF], Theorem 2.11, p.23) was used to derive a sufficient condition for the quenched CLT and its functional form to hold. In order to obtain a similar result under an Orlicz space condition, we will make use of a Rosenthal type inequality for this Orlicz space. Given that no such result seems to exist in the literature, we will follow the outline of the proof given by Burkholder [START_REF] Burkholder | Distribution function inequalities for martingales[END_REF] and adapt it to our framework to establish the required inequality. This paper will be structured as follows: in Section 2, we introduce the notations used throughout our article and we present the main results obtained in this work. In particular, we will split the results into two categories: the first one will aggregate theorems dealing with summations over cubic regions only while the other category will deal with results concerning more general rectangular regions. The proofs of these theorems will appear in Section 3 and in Section 4 we improve on the two applications studied by Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF]. These two examples concern the linear and Volterra random fields satisfying Hannan's condition. They are a common occurrence in the field of financial mathematics and economics. Finally, in Section 5 we give the proof of the Rosenthal type inequality for the Orlicz space mentionned throughout this paper.

Framework and results

In all that follows, we consider a probability space (Ω, F, P) and all the random variables considered thereafter will be real-valued and defined on that probability space. We start by introducing multiple notations that will be used throughout this article: d will be an integer greater than 1, bold characters will designate multi-indexes, denote n := (n 1 , . . . , n d ) and |n| := d i=1 n i for any n ∈ Z d . The set of integers {1, . . . , d} will be denoted by 1, d . In order to define the concept of past trajectory, it is necessary to define an order on

Z d : if u, v ∈ Z d are multi-indexes such that for all k ∈ 1, d , u k ≤ v k , then we will write u ≤ v.

------→).

Before introducing the field we are interested in, we define some transformations on Ω. We let T i : Ω → Ω, i ∈ {1, . . . , d} be invertible measure-preserving commuting transforms on the probability space (Ω, F, P) and we make use of the operators notation (i.e. if U and V are two transformations on Ω, we denote U V := U • V ).

We consider a sigma-field F 0 ⊂ F such that F 0 ⊂ T -i F 0 and a random variable X 0 ∈ L 0 2 where L 0 2 = L 0 2 (Ω, F 0 , P) is the set of all F 0 -measurable square integrable centered random variables. For every n = (n 1 , . . . , n d ) ∈ Z d , set

X n = X 0 • T n , (1)
and

F n = T -n F 0 , (2)
where

T n = T n 1 1 • • • T n d d . As a result, X n is F n -measurable.
Suppose that the family (F k ) k∈Z d is a commuting filtration, that is, for every integrable random variable X, we have

E i [E j [X]] = E i∧j [X]
,

where

E i [X] = E [ X| F i ]
and i ∧ j is the coordinate-wise minimum between i and j.

We recall the notion of ortho-martingale which were introduced by Cairoli [START_REF] Cairoli | Un théorème de convergence pour martingales a indices multiples[END_REF] (see also Khoshnevisan [21]). We say that a random field (D i ) i∈Z d is an ortho-martingale difference field if it is integrable and satisfies the equation E a [D n ] = 0 as long as there exists an integer k ∈ 1, d such that a k < n k .

Then, if M n := 0≤u≤n D u the random field (M n ) n∈N d will be called an ortho-martingale.

Suppose also that the random variable X 0 is regular with respect to the filtration

F, that is E [X 0 |F -∞e i ] = 0 for every i ∈ {1, • • • , d}
, where e i is the multi-index whose i-th coordinate is equal to 1 and the others are equal to 0 and using the convention that ∞ × 0 = 0.

We consider the projection operators defined, for any n ∈ Z d , by

P n = d i=1 (E n -E n-e i ).
For every ω ∈ Ω, we denote by P ω a regular version of the conditional probability given F 0 , that is,

P ω = P ( •| F 0 ) (ω).
Finally we introduce the sum that we will be studying, for every n ∈ (N * ) d ,

S n = n i=1 X i := 1≤i≤n X i ,
and we also set

Sn = S n -R n with R n = d i=1 (-1) i-1 1≤j 1 <•••<j i ≤d E n (j 1 ,••• ,j d ) [S n ] ,
where n (j 1 ,••• ,j d ) is the multi-index obtained by replacing with 0 all the j 1 , • • • , j i -th coordinates of the multi-index n and leaving the rest unchanged.

In dimension d = 1, this reduces to the following expression:

Sn = S n -E [S n |F 0 ] , for n ∈ N * .
This case was investigated by Cuny and Volný in [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF] and therefore, we will always consider d > 1 in the rest of the paper. In dimension d = 2, the definition of Sn reduces down to

Sn,m = S n,m -E [S n,m |F n,0 ] -E [S n,m |F 0,m ] + E [S n,m |F 0,0 ] , for (n, m) ∈ (N * ) 2 .

Functional CLT over squares

Here we present the quenched functional CLT over cubic regions of Z d . These results expand to the functional framework Theorem 4.1, the second part of Corollary 4.3 and Theorem 4.4 (a) obtained in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF]. It is also possible to view these results as an extension to higher dimensions of Theorem 1 established by Cuny and Volný in [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF]. As noted in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF], the proof of these theorems essentially reduces down to a particular case of the proof of the functional central limit theorems over rectangular regions of Z d . The differences in the proofs between the two frameworks will be specified in greater details in Section 3.

Theorem 2.1 Assume that (X n ) n∈Z d is defined by (1) and the filtrations are commuting. Also assume that one of the transformations T i , 1 ≤ i ≤ d is ergodic and in addition

u≥0 P 0 (X u ) 2 < ∞, (3)
Then, for P-almost all ω ∈ Ω, .

1 n d/2 S⌊nt⌋ t∈[0,1] d D ------→ n→∞ (σW t ) t∈[0,1] d under P ω , where σ 2 := E D 2 0 with D 0 = i≥0 P 0 (X i ), (W t ) t∈[0,
In Theorem 2.1, the random centering R ⌊nt⌋ cannot be avoided without additional hypothesis. As a matter of fact, for d = 1, Volný and Woodroofe [START_REF] Volný | An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process[END_REF] constructed an example showing that the CLT for partial sums needs not be quenched. It should also be noticed that, for a stationary ortho-martingale, the existence of a finite second moment is not enough for the validity of a quenched CLT when the summation is taken over rectangles (see Peligrad and Volný [START_REF] Peligrad | Quenched invariance principles for ortho-martingale-like sequences[END_REF]). That being said, the following corollary gives a sufficient condition to get rid of the stochastic centering R n in the previous theorem.

Corollary 2.2 Assume the hypothesis of Theorem 2.1 holds and assume that in addition, for every i ∈ {1, . . . , d}, it holds

1 n d E 0 max 1≤m≤n1 (E m (i) [S m ]) 2 a.s.
------→ n→∞ 0 where we recall that m (i) is the multi-index obtained by replacing with 0 the i-th coordinate of the multi-index m and leaving the rest unchanged. Then, for almost all ω ∈ Ω,

1 n d/2 S ⌊nt⌋ t∈[0,1] d D ------→ n→∞ (σW t ) t∈[0,1] d under P ω , (4)
where (W t ) t∈[0,1] d is a standard Brownian sheet and the convergence happens in the Skhorohod space D([0, 1] d ) endowed with the uniform topology.

To end this section, we give a condition that is easier to verify but still guarantees that the convergence (4) holds.

Theorem 2.3 Assume that u≥1 E 1 (X u ) 2 |u| 1 2 < ∞. (5)
Then, for almost all ω ∈ Ω, the conclusion of Corollary 2.2 holds.

Once again we note that this result is an extension of Corollary 2 in [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF] to random fields and an extension to the functional framework of Theorem 2.6 (a) found in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF].

Functional CLT over rectangles

In order to obtain a functional CLT when we sum over rectangles, a stronger projective condition than (3) is necessary. Indeed, Peligrad and Volný [START_REF] Peligrad | Quenched invariance principles for ortho-martingale-like sequences[END_REF] have given a counter example to a quenched CLT over rectangles for some stationary ortho-martingale under condition (3). This leads us to consider a projective condition in an Orlicz space associated to a specific Young function.

Following the work of Krasnosel'skii and Rutitskii [START_REF] Krasnosel'skii | Convex functions and Orlicz spaces[END_REF], we define the Luxemburg norm associated to the Young function Φ : R + \ {0} → R + \ {0} as

f Φ = inf{t > 0 : E [Φ (|f |/t)] ≤ 1}.
In everything that follows, we will consider the Young function Φ d : R

+ \ {0} → R + \ {0} defined for every x ∈ R + \ {0} by Φ d (x) = x 2 (log(1 + |x|)) d-1 . (6)
Theorem 2.4 Assume that (X n ) n∈Z d is defined by ( 1) and the filtrations are commuting. Also assume that one of the transformations T i , 1 ≤ i ≤ d is ergodic and in addition

u≥0 P 0 (X u ) Φ d < ∞,
where we recall that

Φ d (x) = x 2 (log(1 + |x|)) d-1 . Then, for P-almost all ω ∈ Ω, 1 |n| S[tn] t∈[0,1] d D ------→ n→∞ (W t ) t∈[0,1] d under P ω ,
where [tn] is the integer part of the vector tn :

= (t 1 n 1 , . . . , t d n d ), σ 2 is defined in 2.1, (W t ) t∈[0,1] d is a
Brownian sheet and the convergence happens in the Skhorohod space D([0, 1] d ). In addition,

σ 2 = lim n→∞ E[ S2 n ] |n|
We remark that this result and the ones following extend Theorem 4.2, the first part of Corollary 4.3 and Theorem 2.6 (b) in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] by obtaining a functional version of these theorems.

Corollary 2.5 Assume the hypothesis of the previous theorem hold and assume that in addition, for every i ∈ 1, d ,

1 |n| E 0 max 1≤m≤n (E m (i) [S m ]) 2 a.s.
------→ n→∞ 0.

Then, for P-almost all ω ∈ Ω,

1 |n| S [tn] t∈[0,1] d D ------→ n→∞ (W t ) t∈[0,1] d under P ω , (7)
where (W t ) t∈[0,1] d is a Brownian sheet and the convergence happens in the Skhorohod space D([0, 1] d ).

Theorem 2.6 Assume that the hypothesis of Theorem 2.4 and (5) hold. Then for almost all ω ∈ Ω,

This last Theorem not only extend Theorem 4.4 (b) in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] to the functional case but also reduces the required condition even in the classical CLT case.

Theorem 2.7 Assume that u≥1 E 1 [X u ] Φ d Φ -1 d (|u|) < ∞. (8)
Then, for almost all ω ∈ Ω, the conclusion of Corollary 2.5 holds.

Proofs of the results

Before we prove the previous results, we start by defining some additional notations:

• if h : Ω → R is a measurable function, we will denote by h u , u ∈ Z d , the function h • T u ;
• for any n ∈ (N * ) d and for any measurable function h : Ω → R, we denote

S n (h) = 1≤i≤n h i , and
Sn (h) = S n (h) -R n (h) where R n (h) = d i=1 (-1) i-1 1≤j 1 <•••<j i ≤d E n (j 1 ,••• ,j i ) [S n (h)] ,
and

n (j 1 ,••• ,j i )
is the multi-index whose j 1 , • • • , j i -th coordinates are 0 and the others are equal to the corresponding coordinates of n;

• for any i ∈ 1, d and for any ℓ ∈ N, we denote

F (i) ℓ = k∈Z d k i ≤l F k ; • we set L 2 log d-1 L(G) to be the set of G-measurable functions h : R → R such that E h 2 max(0, log(|h|)) d-1 < ∞; if G = F, we simply write L 2 log d-1 L(F) = L 2 log d-1 L; • if h ∈ L 2 log d-1 L, then we define the maximal operator h * = sup m>0 1 |m| 1≤i≤m |h| • T i .
Let us start with the proof of Theorem 2.4 as it is the most general result. Moreover the computations used in the proof of Theorem 2.1 are a particular case of the computations for the proof of Theorem 2.4 and will be largely skipped.

The proof of Theorem 2.4 relies on the following important lemma which we will refer to as the main lemma in the rest of the paper Lemma 3.1 (Main lemma) For any function h ∈ L 2 log d-1 L, F 0 -measurable and satisfying the following condition:

u≥0 P 0 (h u ) Φ d < ∞, (9)
there exists an integrable function g such that for all N ∈ (N * ) d ,

E 0 max 1≤n≤N 1 |n| |S n (h)| 2 ≤ g P -a.s.
In order to establish this lemma, we shall first obtain the following intermediary lemma.

Lemma 3.2 For any function h ∈ L 2 log d-1 L, there exists a constant C > 0 such that for all u ∈ Z d , we have

|P 0 (h u )| 2 * 1 ≤ C P 0 (h u ) Φ d . Proof of Lemma 3.2. Let h ∈ L log d-1 L, u ∈ Z d and t > P 0 (h u ) Φ d . We let Ω t = ω ∈ Ω : 4 (P 0 (h u )) 2 (ω) > t 2 .
According to Corollary 1.7 of Chapter 6 in Krengel [START_REF] Krengel | Ergodic Theorems[END_REF], there exists a constant C d > 0 such that

P   sup n∈(N * ) d 1 |n| 1≤i≤n P 0 (h u ) • T i 2 > t 2   ≤ C d Ωt 4 (P 0 (h u )) 2 t 2 log 4 (P 0 (h u )) 2 t 2 d-1 dP ≤ 2 d-1 C d Ωt (2P 0 (h u )) 2 t 2 log 1 + 2|P 0 (h u )| t d-1 dP ≤ 2 d+1 C d t -2 P 0 (h u ) 2 Φ d .
The last inequality results from the fact that

P 0 (h u ) Φ d = inf t > 0 : E Φ d |P 0 (h u )| t ≤ 1 .
Indeed, by letting t 0 = P 0 (h u ) Φ d , we have

E (P 0 (h u )) 2 log 1 + |P 0 (h u )| t 0 d-1 ≤ t 2 0 .
Hence since t > t 0 ,

Ω (P 0 (h u )) 2 t 2 log 1 + |P 0 (h u )| t d-1 dP ≤ t 2 0 t -2 ;
Therefore, applying this inequality to

h ′ = 2h ∈ L 2 log d-1 L, we get |P 0 (h u )| 2 * 1 = ∞ 0 P   sup n∈(N * ) d 1 |n| 1≤i≤n P 0 (h u ) • T i 2 > t 2   dt ≤ t 0 0 1dt + ∞ t 0 P   sup n∈(N * ) d 1 |n| 1≤i≤n P 0 (h u ) • T i 2 > t 2   dt ≤ (2 d+1 C d + 1) P 0 (h u ) Φ d .
Proof of the main lemma. We consider a measurable function h satisfying the hypothesis of the lemma and we let n, N ∈ (N * ) d such that n ≤ N . Then, we start by studying the quantity Sn (h) using the following projective decomposition (see [START_REF] Peligrad | Martingale approximation for random fields[END_REF]):

S n (h) -R n (h) = 1≤i≤n P i   i≤u≤n h u   = 1≤i≤n P 0   0≤u≤n-i h u   • T i .
By exchanging the sums, we get

Sn (h) = 0≤u≤n-1 1≤i≤n-u P 0 (h u ) • T i .
Then, recalling that n ≤ N , we obtain

| Sn (h)| ≤ 0≤u≤N -1 max 1≤k≤N 1≤i≤k P 0 (h u ) • T i .
Note that for all u ≥ 0, the partial sum 1≤i≤k P 0 (h u ) • T i is an ortho-martingale. Using Cairoli's inequality for ortho-martingales (see [START_REF] Cairoli | Un théorème de convergence pour martingales a indices multiples[END_REF]), we find that

0≤u≤N -1 E 0   max 1≤k≤N 1≤i≤k P 0 (h u ) • T i 2   ≤ 2 2d u≥0 E 0     1≤i≤N P 0 (h u ) • T i   2   .
By orthogonality, we obtain for all N ∈ (N * ) d

E 0 max 1≤n≤N | Sn (h)| 2 ≤ 2 d u≥0 1≤i≤N E 0 [(P 0 (h u )) 2 • T i ] ≤ 2 d |N | u≥0 |P 0 (h u )| 2 * .
Hence for all N ∈ (N * ) d

E 0 max 1≤n≤N 1 |n| |S n (h)| 2 ≤ 2 d u≥0 |P 0 (h u )| 2 * . ( 10
)
However, according to Lemma 3.2 and hypothesis ( 9), there exists C > 0 such that

u≥0 |P 0 (h u )| 2 * 1 ≤ C u≥0 P 0 (h u ) Φ d < ∞.
This concludes the proof of the main lemma. Proof of Theorem 2.4. For any n ∈ N * , we let

X (n) 0 = j∈ -n,0 d P j (X 0 )
Given the regularity of X 0 , the sequence of random variable (X 0 -X (n) 0 ) n∈N converges almost surely to 0 and using [START_REF] De La Peña | Decoupling : from dependence to independence[END_REF], we get the inequality lim sup

N →∞ E 0 max 1≤m≤N 1 |m| |S m (X 0 -X (n) 0 )| 2 ≤ 2 d u≥0 P 0 (X 0 -X (n) 0 ) • T i 2 *
for all n ∈ N * . Then, according to the proof of the main lemma, there exists a constant C such that lim sup

N →∞ E 0 max 1≤m≤N 1 |m| |S m (X 0 -X (n) 0 )| 2 1 ≤ C u≥0 P 0 (X 0 -X (n) 0 ) • T u Φ d ------→ n→∞ 0.
Therefore, there exists an increasing sequence of integers (n k ) k∈N such that lim k→∞ lim sup

N →∞ E 0 max 1≤m≤N 1 |m| |S m (X 0 -X (n k ) 0 )| 2 = 0 a.s. ( 11
)
Moreover, we also have, for all n ∈ N *

1 |N | E 0 max 1≤i≤N |R i (X (n) 0 )| 2 a.s.
------→

N →∞ 0. ( 12
)
Indeed, using the triangular inequality, it is enough to show that for all i ∈ -n,

0 d 1 |N | E 0 max 1≤j≤N |R j (P i (X 0 ))| 2 a.s.
------→

N →∞ 0.
This holds true by applying following lemma.

Lemma 3.3 For any square integrable function F 0 -measurable h, the condition

u≥1 E 1 [h u ] 2 |u| 1/2 < ∞ (13) implies 1 |N | E 0 max 1≤n≤N |R n (h)| 2 a.s.
------→

N →∞ 0.
We delay the proof of this lemma to latter in this section.

Remark that the proof of Proposition 4.1 in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] can be easily adapted to the case of Orlicz spaces such that for some fixed n ∈ N * , we get the following martingale-codoundary decomposition

X (n) 0 = S⊂ 1,d h (n) S • j∈S c (I -T j ),
where h

(n) S ∈ i∈S L 2 log d-1 L F (i) 0 ⊖ L 2 log d-1 L F (i) -1
for all S ⊂ 1, d and using the convention j∈∅ (I -T j ) = I. Moreover

h (n) 1,d = i∈Z d P 0 X (n) 0 • T i .
According to the proof of Remark 11 in [START_REF] Peligrad | Quenched invariance principles for ortho-martingale-like sequences[END_REF] (see also the proof of Theorem 7 in the same article), the following almost-sure convergence

P ω max 1≤m≤N 1 |m| S m X (n) 0 -d n 2 ≥ ǫ a.s. ------→ N →∞ 0 (14)
holds for all ǫ > 0, where d n = h

S N (D 0 -d n ) = 1≤i≤N D i -D (n) i
,

where D i = j∈Z d P i (X i-j ) and D (n) i = j∈ -n,0 d P i (X i-j ).
Hence, given that D i -D

(n) i i∈Z d is an ortho-martingale difference field and according to Cairoli's inequality, we have

P ω 1 |N | max 1≤i≤N |S i (D 0 -d n )| > ǫ ≤ 2 2d ǫ 2 |N | 1≤i≤N E 0 D i -D (n) i 2 .
Let us note that

1 |N | 1≤i≤N E 0 D i -D (n) i 2 ≤ j ∈ -n,0 d 1 |N | 1≤i≤N E 0 (P 0 (X -j )) 2 • T i .
According to the ergodic Theorem 1.1 of Chapter 6 in Krengel [START_REF] Krengel | Ergodic Theorems[END_REF] for Dunford Schwartz operators and Lemma 7.1 in [START_REF] Dedecker | A quenched weak invariance principle[END_REF], we have the following convergence :

lim N →∞ 1 |N | 1≤i≤N E 0 (P 0 (X -j )) 2 • T i = E (P j (X 0 )) 2 a.s.
for all j ∈ -n, 0 d . Since j∈≥0 P 0 (X j ) 2 < ∞, we get

lim n→∞ lim N →∞ 2 2d ǫ 2 |N | 1≤i≤N E 0 D i -D (n) i 2 = 0 a.s. (15)
Combining ( 11), ( 12),( 14) and ( 15), we obtain that for all ǫ > 0, lim sup

N →∞ P ω 1 |N | max 1≤m≤N | Sm -S m (D 0 ) | ≥ ǫ = 0 a.s.
We conclude by noticing that the field (D 0 • T i ) i∈Z d satisfies a functional central limit theorem (according to Theorem 5 in [START_REF] Peligrad | Quenched invariance principles for ortho-martingale-like sequences[END_REF]) and therefore the expected result is obtained by applying Theorem 3.1 in [START_REF] Neuhaus | On weak convergence of stochastic processes with multidimensional time parameter[END_REF].

Proof of the Theorem 2.1. The proof of this theorem is very similar to the previous one, with the exception of using Theorem 2.8 instead of Theorem 1.1 of Chapter 6 in Krengel [START_REF] Krengel | Ergodic Theorems[END_REF] and Lemma 1.4 in the same Chapter (applied to the abstract maximal operator M f := sup n∈N

1 n d 1≤i≤n1 |f | • T i , see Definition 1.
3 of Chapter 6 and Corollary 2.2 of Chapter 1 in Krengel [START_REF] Krengel | Ergodic Theorems[END_REF]) instead of Corollary 1.7 in order to obtain the L 2 versions of intermediary lemma and the main lemma mentioned below. Lemma 3.4 (Version L 2 ) For any function h ∈ L 2 (F 0 ) satisfying the following condition :

u≥0 P 0 (h u ) 2 < ∞, (16)
there exist an integrable function g such that for all N ∈ N * ,

E 0 max 1≤n≤N 1 n d |S n1 (h)| 2 ≤ g P -a.s.
Lemma 3.5 (Version L 2 of Lemma 3.2) For all functions h ∈ L 2 , there exists a constant C > 0 such that for all u ∈ Z d , we have

|P 0 (h u )| 2 ⋆ 1 ≤ C P 0 (h u ) 2 ,
where

h ⋆ = sup n∈N * 1 n d 1≤i≤n1 |h| • T i .
The following proof of Corollary 2.5 can be easily adapted to obtain Corollary 2.2 by using the Theorem 2.1 instead of Theorem 2.4.

Proof of Corollary 2.5.

According to Theorem 2.4 and Theorem 3.1 in [START_REF] Neuhaus | On weak convergence of stochastic processes with multidimensional time parameter[END_REF], it is enough to show

1 |n| E 0 max 1≤i≤n R 2 i a.s.
------→ n→∞ 0.

Let n ∈ Z d , and recall that

R n = d i=1 (-1) i-1 1≤j 1 <•••<j i ≤d E n (j 1 ,••• ,j i ) [S n ] ,
where n (j 1 ,...,j i ) is the multi-index such that the j k -th, 1 ≤ k ≤ i coordinates are zero and the other are equal to the corresponding coordinates of n.

Using the triangular inequality, it is enough to prove that the property

1 |n| E 0 max 1≤i≤n E n (j 1 ,••• ,j i ) [S n ] 2 a.s.
------→ n→∞ 0 P (j 1 , . . . , j i ) holds for any

j 1 < • • • < j i , 1 ≤ i ≤ d.
We establish that via induction on i.

The terms satisfying i = 1 have this property according to the hypothesis of the corollary. By induction, the corollary will be proven if we can show that if property P (j 1 , . . . , j i ) is verified for all j 1 < • • • < j i for some i < d, then P (j 1 , . . . , j i+1 ) also holds true for all j 1 < • • • < j i+1 . For the sake of simplicity and without loss of generality, we will only establish that ∀j ∈ 1, d , P (j) =⇒ P (1, 2).

In other words, we use the hypothesis of the corollary to show

1 |n| E 0 [ max 1≤i≤n (E i (1,2) [S i ]) 2 ]
a.s.

------→ n→∞ 0.

According to Jensen's inequality we have

E 0 [ max 1≤i≤n (E i (1,2) [S i ]) 2 ] = E 0 [ max 1≤i≤n (E i (1,2) [E i (1) [S i ]]) 2 ] ≤ E 0 [ max 1≤i≤n (E i (1) [S i ]) 2 ]. So 1 |n| E 0 [ max 1≤i≤n (E i (1,2) [S i ]) 2 ] ≤ 1 |n| E 0 [ max 1≤i≤n (E i (1) [S i ]) 2 ]
a.s.

------→ n→∞ 0.

Before continuing with the proof, we establish Lemma 3.3.

Proof of Lemma 3.3. We show that for any i ∈ 1, d and any 1 ------→ N →∞ 0.

≤ j 1 < • • • < j i ≤ d, we have the convergence 1 |N | E 0 max 1≤n≤N E n (j 1 ,...,j i ) [S n (h)]
In order to do so, we use an induction on k = d -i with d fixed. For k = 0 and in the same way as in the proof of Lemma 3.2 in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF], we establish that

(E 0 [S n (h)]) 2 |n| a.s. ------→ n→∞ 0. Hence 1 |N | max 1≤n≤N (E 0 [S n (h)]) 2 a.s. ------→ N →∞ 0.
Now, we suppose that the desired property holds true for some k-1 < d-1. Without loss of generality, we establish the property only for (j 1 , . . . , j n ) = (1, . . . , n); it is enough to show that ------→

1 |N | E 0 max 1≤n≤N E n (j 1 ,...,j k ) [S n (h)] -E n (j 2 ,...,j k ) [S n (h)]
N →∞ 0.
Let n, N ∈ (N * ) d such that n ≤ N then the following decomposition holds (see the proof of Lemma 3.2 in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF])

E 0 max 1≤n≤N E n (j 1 ,...,j k ) [S n (h)] -E n (j 2 ,...,j k ) [S n (h)] 2 = E 0   max 1≤n≤N n 1 i=1 P (1) n (j 1 ,...,j k ) ,i (S n (h)) 2   ,
where P

n (j 1 ,...,j k ) ,i (S n (h)) = E ie 1 +n (j 1 ,...,j k ) [S n (h)] -E (i-1)e 1 +n (j 1 ,...,j k ) [S n (h)] and e 1 is the multi-index whose coordinates are all zero expect for the first one which is 1.

Since h is F 0 -measurable, we have

n 1 i=1 P (1) n (j 1 ,...,j k ) ,i (S n (h)) ≤ 1≤u≤N max 1≤k≤N 1 k i=1 P (1) u (j 1 ,...,j k ) ,i (h u ) .
Therefore, according to Doob's inequality for martingales, it follows

E 0   max 1≤n≤N n 1 i=1 P (1) n (j 1 ,...,j k ) ,i (S n (h)) 2   ≤ 2 1≤u≤N E 0   N 1 i=1 P (1) u (j 1 ,...,j k ) ,i (h u ) 2   .
Let c > 0, we use the following decomposition

1 |N | 1≤u≤N E 0   N 1 i=1 P (1)
u (j 1 ,...,j k ) ,i (h u ) 2   =: I N ,c + II N ,c
where

I N ,c = 1 |N | 1≤u 1 ≤N 1 1≤u 2 ,...u d ≤c E 0   N 1 i=1 P (1)
u (j 1 ,...,j k ) ,i (h u ) 2 
 and II N ,c is the remainder of the initial sum.

Let us show that lim sup

N →∞ I N ,c = 0 a.s.
Indeed, by an orthogonality argument

I N ,c ≤ c d-1 |N | sup 1≤u 2 ,...,u d ≤c u 1 ≥0 N 1 i=1 E 0 P (1)
u (j 1 ,...,j k ) ,0 (h u ) 2 • T i 1 .
However, according to Theorem 1.1 of Chapter 6 in [START_REF] Krengel | Ergodic Theorems[END_REF] and Lemma 7.1 in [START_REF] Dedecker | A quenched weak invariance principle[END_REF], we get the convergence lim

N 1 →∞ 1 N 1 1≤i≤N 1 E 0 P (1)
u (j 1 ,...,j k ) ,0 (h u ) 2 • T i 1 = E P (1) u (j 1 ,...,j k ) ,0 (h u ) 2 |I 1 a.s.
where I 1 is the invariant σ-algebra associated to T 1 . Hence lim sup

N →∞ I N ,c = 0 a.s.
The different sums appearing in II N ,c , admit at least one direction (different from the first one) for which the indices in the sums for this direction are at least equal to c + 1. Without loss of generality, we suppose that it is the second one. Then

1 |N | 1≤u 1 ≤N 1 c+1≤u 2 ≤N 2 1≤u 3 ≤N 3 • • • 1≤u d ≤N d E 0   N 1 i=1 P (1) u (j 1 ,...,j k ) ,i (h u ) 2   ≤ u 2 ≥c+1 u 1 ≥0 u 3 ≥1 • • • u d ≥1 (u 2 • • • u d ) -1 2 1 N 1 N 1 i=1 E 0 P (1)
u (j 1 ,...,j k ) ,0 (h u ) 2 • T i 1 .
Applying, once again Theorem 1.1 of Chapter 6 in [START_REF] Krengel | Ergodic Theorems[END_REF] and Lemma 7.1 in [START_REF] Dedecker | A quenched weak invariance principle[END_REF], we obtain lim sup

N →∞ II N ,c ≤ u 2 ≥c+1 u 1 ≥0 u 2 ≥1 • • • u d ≥1 E P (1) u (j 1 ,...,j k ) ,0 (h u ) 2 |I 1 √ u 2 • • • u d .
Since ( 13) implies (45) in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] (see the proof of Theorem 4.4), we get lim c→∞ lim sup

N →∞ II N ,c = 0 a.s.
This concludes the proof.

Theorems 2.6 and 2.3 are direct consequences of this lemma and the previous results. Proof of Theorem 2.6. This theorem is a consequence of Lemma 3.3 and of Corollary 2.5.

Proof of Theorem 2.3. The theorem is a consequence of Lemma 3.3, (47) in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] (for q = 2; and the proof of Theorem 4.2 and Lemma 3.3 in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF]) and Theorem 2.1.

The rest of this section will be dedicated to proving Theorem 2.7. We start by making a few remarks concerning the Luxemburg norms. Let us note that for x ≥ 0 and 0 < λ ≤ e -1,

log 1 + x λ log(1 + λ) ≤ log(1 + x).
Recall that the function Φ d : R

+ \ {0} → R + \ {0} is defined by Φ d (x) = x 2 (log(1 + |x|)) d-1
for all x ∈ R + \ {0}. Then, we deduce the following remarkable property of the function Φ d . For x > 0 and 0 < λ ≤ e -1,

Φ d x λ = x λ 2 log 1 + x λ d-1 ≤ x 2 (log(1 + x)) d-1 λ 2 (log(1 + λ)) d-1 = Φ d (x) Φ d (λ) . ( 17
)
Besides, since Φ d is a convex function, we also have

Φ d x λ = Φ d x λ + 1 - 1 λ • 0 ≤ Φ d (x) λ + 1 - 1 λ Φ d (0) = Φ d (x) λ , (18)
for x ≥ 0 and λ ≥ 1.

Obviously the function Φ d defined by ( 6) is bijective and we denote by Φ -1 d its inverse function. The following lemma might be well-known but we could not find it in the literature.

Lemma 3.6 Let X ∈ L 2 log d-1 L. If Φ -1 d (E [Φ d (X)]) ≤ e -1, then X Φ d ≤ Φ -1 d (E[Φ d (X)]) , and if E[Φ d (X)] ≥ 1, then X Φ d ≤ E [Φ d (X)] .
Proof of Lemma 3.6. If X = 0 almost surely, then the property is evident. Else, suppose that P(X = 0) = 1, then by the definition of Luxembourg norm

X Φ d = inf λ > 0 : E Φ d X λ ≤ 1 .
Note that by the properties of Φ d for any 0

< λ ≤ e -1 E Φ d X λ ≤ E [Φ d (X)] Φ d (λ) .
From this inequality it follows that if λ is the solution to the equation E

[Φ d (X)] = Φ d (λ) we have necessarily that E Φ d X λ
≤ 1, and then

X Φ d ≤ λ = Φ -1 d (E [Φ d (X)]).
For the case E [Φ d (X)] > 1, the proof is similar using property (18) of Φ d .

Lemma 3.7 Condition (8) implies u≥0 P 0 (X u ) Φ d < ∞.
Proof of Lemma 3. for x ∈ R + \ {0}. By the generalized Holder inequality for Orlicz spaces (see [START_REF] Rao | Theory of Orlicz spaces[END_REF], p.58) we have

u≥1 P 0 (X u ) Φ d = n≥0 2 n+1 -1 v=2 n P 0 (X v ) Φ d ≤2 n≥0 inf    η > 0 : 2 n+1 -1 v=2 n Ψ d 1 η ≤ 1    • inf    η > 0 : 2 n+1 -1 v=2 n Φ d P 0 (X v ) Φ d η ≤ 1    .
We have that inf    η > 0 :

2 n+1 -1 v=2 n Ψ d 1 η ≤ 1    = 1 Ψ -1 d (2 -n ) , where 2 -n = (2 n 1 • • • 2 n d ) -1 .
In order to control the second term in the previous inequality, we make the following remark: if f is a convex function then for any η > 0, the following holds

f   2 n+1 -1 v=2 n P -v (X 0 ) η Φ d   = f   d i=0 (-1) i 1≤j 1 <•••<j i ≤d E -2 n+1 (j 1 ,...,j i ) [X 0 ] Φ d   ≤ 1 2 d d i=0 1≤j 1 <•••<j i ≤d f 2 d η (-1) i E -2 n+1 (j 1 ,...,j i ) [X 0 ] Φ d ≤ f 2 d η E -2 n [X 0 ] Φ d .
It is enough to control the second term in the sum only when E Φ d

P -v (X 0 ) η ≤ Φ d (e -1)
for any v ≥ 0, as the other cases can be proved using similar arguments and are left to the reader. By Lemma 3.6 above, if E Φ d

P -v (X 0 ) η ≤ Φ d (e -1)
for any v ≥ 0, we also have

Φ d P -v (X 0 ) Φ d η ≤ E Φ d P -v (X 0 ) η So 2 n+1 -1 v=2 n Φ d P -v (X 0 ) Φ d η ≤ 2 n+1 -1 v=2 n E Φ d P -v (X 0 ) η . (19)
Before proceeding, we state the following lemma which consist in a version of the Rosenthal inequality in the Orlicz space associated to Φ d . Its proof will be given in the appendix.

Lemma 3.8 If (d u ) u∈(N) * is an ortho-martingale difference field, then there exists two constants

C 1 , C 2 > 1, which only depend on d, such that n-1 u=0 E [Φ d (d u )] ≤ C 1 max   f d • ϕ -1 d   C 2 n-1 u=0 d u Φ d   , φ d   C 2 n-1 u=0 d u Φ d     , with φ d (x) = x d+1 (log(1 + x)) d-1 , ϕ d (x) = x (log(1 + x)) d-1 and f d (x) = x 3-d d-1 (log(1 + x 1 d-1 )) d-1 for all x > 0. If d ≥ 3, then both functions φ d and f d • Ψ -1
d are convex on R * + and so the function ρ : R

+ → R + defined by ρ(x) = max(f d • Ψ -1 d (x), φ d (x)
) is convex too. Using [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF], applying Lemma 3.8 and using the convexity of ρ, we show that there exists C 1 , C 2 > 1 such that for any η > 0,

2 n+1 -1 v=2 n Φ d P 0 (X v ) Φ d η ≤ 2 n+1 -1 v=2 n E Φ d P -v (X 0 ) η ≤ C 1 max   f d • ϕ -1 d   C 2 n-1 u=0 P -v (X 0 ) η Φ d   , φ d   C 2 n-1 u=0 P -v (X 0 ) η Φ d     = C 1 ρ   C 2 n-1 u=0 P -v (X 0 ) η Φ d   .
According to the remark above, we get that

ρ   C 2 n-1 u=0 P -v (X 0 ) η Φ d   ≤ ρ 2 d C 2 η E -2 n [X 0 ] Φ d .
However

C 1 ρ C 2 2 d η E -2 n [X 0 ] Φ d ≤ 1 ⇐⇒ 2 d C 2 φ -1 d (C -1 1 ) E -2 n [X 0 ] Φ d ≤ η ≤ 2 d C 2 (f d • ϕ -1 d ) -1 (C -1 1 ) E -2 n [X 0 ] Φ d . Therefore, setting C = 2 d C 2 φ -1 d (C -1 1 ) , we conclude that inf    η > 0 : 2 n+1 -1 v=2 n Φ d P 0 (X v ) Φ d η ≤ 1    ≤ inf η > 0 : C 1 ρ C 2 2 d η E -2 n [X 0 ] Φ d ≤ 1 = C E -2 n [X 0 ] Φ d .
If d = 2, then we set ρ(x) = max(x, φ 2 (x)) for all x ≥ 0 and a similar reasoning leads to the same inequality. Since E -n [X 0 ] Φ d is non-increasing in all directions of n, we obtain that for any n such that n k > 0 for all k ∈ 1, d , we have

2 n Φ d -1 (2 n ) E -2 n [X 0 ] Φ d ≤ 2 n -1 u=2 n-1 E 0 [X u ] Φ d Φ d -1 (|u|) .
So, for some positive constant

K v≥1 P 0 (X u ) Φ d ≤ K n≥1 Φ -1 d (2 n ) 2 n Ψ d -1 (2 -n ) 2 n -1 u=2 n-1 E -u (X 0 ) Φ d Φ d -1 (|u|)
However, there exists a constant

K ′ > 0 such that Φ d -1 (2 n ) ∼ n→∞ 2 n Ψ d -1 (2 -n )K ′ . Hence, according
to the previous inequalities, we have shown that (8) implies

u≥1 P 0 (X u ) Φ d < ∞.
In the same way, we have for every i ∈ 1, d and for every (j 1 , . . . , j i ) ∈ 1, d i such that j 1 < • • • < j i , u (j 1 ,...,j i ) ≥1 (j 1 ,...,j i ) P 0 (X u (j 1 ,...,j i ) ) Φ d < ∞.

Hence (2.4) is fullfilled.

Proof of Theorem 2.7. This theorem is a consequence of Theorem 2.4 and Lemmas 3.3 and 3.7.

Examples

In this section, we present two examples of application of the previous results we obtained. We will focus on both a linear and a non-linear example. We improve on the results by Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] by requiring weaker assumptions on both the moment of the innovations and the coefficients which appear in each example. More precisely, we obtain a functional CLT despite only requiring that the i.i.d. innovations belongs to the Orlicz space L 2 log d-1 L instead of the Lebesgue space L q with q > 2 as is required by Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF]. Throughout this section, as before, we denote by C > 0 a generic constant, which may be different from line to line. Recall that the function Φ d : R + \ {0} → R + \ {0} is bijective and defined by [START_REF] Cuny | On martingale approximations and the quenched weak invariance principle[END_REF].

Example 4.1 (Linear field) Let (ξ n ) n∈Z d be a random field of independent, identically distributed random variables, which are centered and satisfy

E |ξ 0 | 2 (log(1 + |ξ 0 |)) d-1 < ∞. For k ≥ 0 define X k = j≥0 a j ξ k-j .
Assume that the following condition hold.

k≥1 1 Φ d -1 (|k|) j≥k-1 a 2 j 1 2 < ∞. ( 20
)
Then the quenched functional CLT in Corollary 2.5 holds.

Remark 4.2 Condition (20) is fulfilled whenever the stronger but more practical condition

k≥1 (log(|k|)) 1/2 |k| 1/2 j≥k-1 a 2 j 1 2 < ∞ is satisfied.
The results obtained by Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] (Remark 6.2 (c)) required the existence of q-th moment, with q > 2, of the innovation ξ 0 to obtain the quenched functional CLT;, meanwhile, we only require that ξ 0 satisfy a weaker Orlicz condition to obtain that result. Additionally we require weaker assumptions on the coefficients a u , u ∈ Z d . Proof of Example 4.1. According to the independence of ξ n , we have

E 1 [X u ] = j≥u-1 a j ξ k-j .
Then, applying the Burkholder inequality for Orlicz spaces (see Meyer and Dellacherie [START_REF] Dellacherie | Probabilités et potentiel: Chapitres V à VIII: Théorie des martingales[END_REF], p.304, VII -92), we obtain

E 1 [X u ] Φ d = j≥u-1 a j ξ k-j Φ d ≤ C   j≥u-1 a 2 j ξ k-j 2 Φ d   1 2
By stationarity and since ξ 0 Φ d < ∞, we obtain, by using assumption [START_REF] Kipnis | Central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions[END_REF], that condition (8) is satisfied. Hence the result.

Example 4.3 (Volterra field) Let (ξ n ) n∈Z d be a random field of independent, identically distributed, and centered random variables satisfying

E |ξ 0 | 2 (log(1 + |ξ 0 |)) d-1 < ∞. For k ≥ 0, define X k = (u,v)≥(0,0) a u,v ξ k-u ξ k-v .
where a u,v are real coefficients with a u,u = 0 and u,v≥0 a 2 u,v < ∞. In addition, assume that it holds that

k≥1 1 Φ d -1 (|k|) (u,v)≥(k-1,k-1) u =v a 2 u,v 1/2 < ∞, (21)
Then the quenched functional CLT in Theorem 2.5 holds.

Remark 4.4 In the same manner as before, we notice that (21) is satisfied whenever

k≥1 (log(|k|)) 1/2 |k| 1/2 (u,v)≥(k-1,k-1) u =v a 2 u,v 1/2 < ∞.
This is a generalization of the quenched functional CLT obtained in [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] (Example 6.3). Here, we only require an Orlicz space type condition on the innovation ξ 0 and we weaken the condition (54) in Zhang et al. [START_REF] Zhang | On the quenched CLT for stationary random fields under projective criteria[END_REF] to condition [START_REF] Khoshnevisan | Multiparameter processes. An introduction to random fields[END_REF].

Proof of Example 4.3. Note that

E 1 [X k ] = u,v≥k-1 a u,v ξ k-u ξ k-v
Let (ξ ′ n ) n∈Z d and (ξ ′′ n ) n∈Z d be two independent copies of (ξ n ) n∈Z d . By applying the decoupling inequality (see Theorem 3.1.1 in De la Peña and Giné [START_REF] De La Peña | Decoupling : from dependence to independence[END_REF], p.99) and Jensen's inequality, we get for any t > 0,

E [Φ d (|E 1 [X k ]| /t)] = E   Φ d   1 t u,v≥k-1 a u,v ξ k-u ξ k-v     ≤ E   Φ d   C t u,v≥k-1 a u,v ξ ′ k-u ξ ′′ k-v     . Hence E 1 [X k ] Φ d ≤ C u,v≥k-1 a u,v ξ ′ k-u ξ ′′ k-v Φ d .
Therefore, applying the Burkholder inequality for Orlicz spaces (see Meyer and Dellacherie [START_REF] Dellacherie | Probabilités et potentiel: Chapitres V à VIII: Théorie des martingales[END_REF], p.304, VII -92), we get

E 1 [X k ] Φ d ≤ C     (u,v)≥(k-1,k-1) u =v a 2 u,v ξ k-u Φ d ξ k-v Φ d     1 2
.

By stationarity and since ξ 0 Φ d < ∞, we obtain, by using assumption [START_REF] Khoshnevisan | Multiparameter processes. An introduction to random fields[END_REF], that condition (8) holds. Thus the CLT in Corollary 2.5 holds.

Appendix

In this section we give the proof of Lemma 3.8. We will follow the outline of the proof given by Burkholder [START_REF] Burkholder | Distribution function inequalities for martingales[END_REF] for the Rosenthal inequality in Lebesgue spaces but first, we need to establish a preliminary lemma concerning the Orlicz norm studied in this document. We start by recalling the definition of the different tools we will require.

Recall that the Luxemburg norm associated to the the Young function Φ

d : x ∈ R + \ {0} → Φ d (x) = x 2 (log(1 + |x|)) d-1 ∈ R + \ {0} is defined as f Φ = inf{t > 0 : E [Φ (|f |/t)] ≤ 1},
and by Ψ d we denote the conjugate function associated with Φ d defined in the following way

Ψ d (x) = sup y≥0 (xy -Φ d (y)).
Besides properties ( 17) and ( 18), the natural logarithm also satisfies

log 1 + x λ log(1 + λ) ≥ log(2)λ log (1 + x) , (22)
for all 0 < λ < 1 and x ≥ 0 as well as

log 1 + x λ log(1 + λ) ≥ log(2) log (1 + x) λ , (23)
all λ ≥ 1 and x ≥ 0. The following lemma will help us compute the Orlicz norm associated to Ψ d of a specific random variable which will appear in the proof of Lemma 3.8.

Lemma 5.1 For h ∈ L 2 log L taking nonnegative values, we have

h (log(1 + h)) d-1 Ψ d ≤ h Φ d .
Proof of Lemma 5.1. Let h ∈ L 2 log L and t > 0. Using the inequality Ψ(x(log(1 + x)) d-1 /t) ≤ Φ(x/t) for all x ≥ 0, we get

E Ψ d h log d-1 (1 + h) t ≤ E Φ d h t . Taking t = h Φ d , we obtain h (log(1 + h)) d-1 Ψ d ≤ h Φ d .
We can now prove Lemma 3.8. In order to do so, we will make use of Lemma 3.1 in [START_REF] Burkholder | Distribution function inequalities for martingales[END_REF].

Proof of Lemma 3.8. We start by introducing a few items of notation. For all n ∈ N d , we denote M n = n-1 u=0 d u and σ n = n-1 u=0 d 2 u . Our proof will be decomposed into two parts. In the first part, we will make the additional assumption that the ortho-martingale (M n ) n is nonnegative. Then in the second part, we will establish the result for real-valued ortho-martingales.

First step: We suppose that (M n ) n∈(N * ) d is a nonnegative ortho-martingale. Let n ∈ N d be fixed and remark that, since Φ

d ( √ a + b) ≥ Φ d ( √ a) + Φ d ( √ b) for all a, b ≥ 0, it holds that E Φ d σ n η ≥ n-1 u=0 E Φ d d u η . (24)
for any η > 0. Let X = max(σ n , max 0≤u≤n M u ) and suppose that X Φ d ≤ 1. Applying [START_REF] Klicnarová | Limit theorems for weighted Bernoulli random fields under Hannan's condition[END_REF] with η = 1, we get

σ n Φ d ≤ X Φ d ≤ 1 and η 0 := n-1 u=0 E [Φ d (d u )] ≤ 1 (25) Setting f d (x) = x 3-d d-1 log 1 + x 1 d-1 d-1
for all x > 0 and η ′ 0 = η 0 log(2) d-1 , we find that

E   Φ d   σ n f -1 d (η ′ 0 ) 1 d-1     ≥ 1 (f -1 d (η ′ 0 )) 2 d-1 E   σ 2 n   log   1 + σ n f -1 d (η ′ 0 ) 1 d-1     d-1    ≥ 1 η 0 E [Φ d (σ n )] ≥ 1.
The second to last inequality holds since according to [START_REF] Krasnosel'skii | Convex functions and Orlicz spaces[END_REF], we have

  log   1 + σ n f -1 d (η ′ 0 ) 1 d-1     d-1 ≥ f -1 d (η ′ 0 ) log(2) d-1 (log (1 + σ n )) d-1 log 1 + f -1 d (η ′ 0 ) 1 d-1 d-1 = (f -1 d (η ′ 0 )) 2 d-1 log(2) d-1 (log (1 + σ n )) d-1 f d (f -1 d (η ′ 0 )) = (f -1 d (η ′ 0 )) 2 d-1 η 0 (log (1 + σ n )) d-1 .
From the previous inequality, we deduce that Computing

√ 3X 0 Φ ′ d (u)
u du, we find that

√ 3X 0 Φ ′ d (u) u du ≤ 3 √ 3X log 1 + √ 3X d-1 . Thus E [Φ d (X)] ≤ 3 d+5 2 Ω M n X (log (1 + X)) d-1 dP (27)
Applying Holder's inequality for Orlicz spaces, we get

Ω M n X (log (1 + X)) d-1 dP ≤ 2 M n Φ d X (log (1 + X)) d-1 Ψ d .
Using Lemma 5.1, we find that

X (log (1 + X)) d-1 Ψ d ≤ X Φ d Then Ω M n X (log (1 + X)) d-1 dP ≤ 2 M n Φ d X Φ d .
Recalling [START_REF] Peligrad | Quenched Invariance Principles via Martingale Approximation; in Asymptotic laws and methods in stochastics[END_REF], we deduce that

E [Φ d (X)] ≤ 2 • 3 d+5 2 M n Φ d X Φ d
Thus, recalling that X Φ d ≤ 1 and applying Lemma 3.6, we obtain

ϕ d X Φ d ≤ 2 • 3 d+5 2 M n Φ d , (28)
with ϕ d (x) = x (log(1 + x)) d-1 for all x ≥ 0. Keeping in mind inequality [START_REF] Neuhaus | On weak convergence of stochastic processes with multidimensional time parameter[END_REF], ( 26) and ( 28), we obtain

n-1 u=0 E [Φ d (d u )] ≤ log(2) 1-d f d • ϕ -1 d 2 • 3 d+5 2 M n Φ d . (29)
Now, suppose that X Φ d > 1. According to the previous case, we have

n-1 u=0 E Φ d d u X Φ d ≤ log(2) 1-d f d (1) = 1.
For any 0 ≤ u ≤ n -1 and by making use of inequality [START_REF] Krengel | Ergodic Theorems[END_REF], it holds that

E Φ d d u X Φ d = E   d 2 u X 2 Φ d log 1 + |d u | X Φ d d-1   ≥ E    log(2) d-1 Φ d (d u ) X d+1 Φ d log 1 + X Φ d d-1    .
We conclude that

) 30
where φ d (x) = x d+1 (log(1 + x)) d-1 , for all x ≥ 0. Once again, by the same argument as in the first case, we get

E [Φ d (X)] ≤ 2 • 3 d+5 2 M n Φ d X (log (1 + X)) d-1 Ψ d .
However using Lemma 5.1, we obtain

X (log (1 + X)) d-1 Ψ d ≤ X Φ d ,
Since log is an increasing function and X Φ d > 1, we deduce that

1 = E Φ d X X Φ d ≤ E [Φ d (X)] X 2 Φ d
and so

X 2 Φ d ≤ E [Φ d (X)] ≤ 2 • 3 d+5 2 M n Φ d X Φ d . Thus X Φ d ≤ 2 • 3 d+5 2 M n Φ d . (31)
Combining [START_REF] Rao | Theory of Orlicz spaces[END_REF] and [START_REF] Volný | A central limit theorem for fields of martingale differences[END_REF], we get the following inequality.

n-1 u=0 E [Φ d (d u )] ≤ log(2) 1-d φ d 2 • 3 d+5 2 M n Φ d . (32)
Finally, recalling [START_REF] Peligrad | Quenched invariance principles for ortho-martingale-like sequences[END_REF] and [START_REF] Volný | Martingale-coboundary decomposition for stationary random fields[END_REF], there exists C 1 , C 2 > 1 only depending on d such that

n-1 u=0 E [Φ d (d u )] ≤ C 1 max f d • ϕ -1 d C 2 M n Φ d , φ d C 2 M n Φ d .
Second step: Now suppose that M can take nonpositive values. We let

M + u = E [
E Φ d (d u ) - ≤ 6 C 1 max f d • ϕ -1 d C 2 M + n Φ d , φ d C 2 M + n Φ d + C 1 max f d • ϕ -1 d C 2 M - n Φ d , φ d C 2 M - n Φ d ≤ 12C 1 max f d • ϕ -1 d C 2 M n Φ d , φ d C 2 M n Φ d .
The proof of the theorem is complete.

  Convergence of fields indexed by Z d will be interpreted in the following sense. If n = (n 1 , . . . , n d ) is a multi-index, then the notation n → ∞ is to be interpreted as the convergence of min{n 1 , . . . , n d } to ∞. Convergence in distribution (resp. almost surely) will be denoted by D ------→ (resp. a.s.

  . Moreover, letting N ∈ (N * ) d and D 0 = i∈Z d P 0 (X i ), we get

7 .

 7 Let a, b ∈ Z d such that a ≤ b. Denote by Ψ d the conjugate function associated with Φ d defined in the following way Ψ d (x) = sup y≥0 (xy -Φ d (y))

log( 2 ) 1 u=0E

 21 1-d f d σ n Φ d ≥ η 0 = n-[Φ d (d u )] .

n- 1 u=0Elog 1 + X Φ d d- 1 =

 111 [Φ d (d u )] ≤ log(2) 1-d X d+1 Φ d : log(2) 1-d φ d ( X Φ d ).(

2 . 1 u=0E- 1 u=0E Φ d d - u ≤ C 1 max f d • ϕ - 1 dC 2 1 u=0E 6 n- 1 u=0E

 211u12161 max(M n , 0) |G u ] and M - u = E [max(-M n , 0) |G u ] , with 0 ≤ u ≤ n and G u = σ (M v , v ≤ u). Both M +u and M - u are ortho-martingales and satisfy the conditions of the first part. Let n ∈ (N * ) d , we define Therefore there exists C 1 , C 2 > 1 only depending on d such thatn-Φ d d + u ≤ C 1 max f d • ϕ -1 d C 2 M + n Φ d , φ d C 2 M + n Φ d and nM - n Φ d , φ d C 2 M - n Φ d . Using the inequalities Φ d (a + b) ≤ 2 (Φ d (a) + Φ d (b) + 2 max (Φ d (a), Φ d (b))) ≤ 6 (Φ d (a) + Φ d (b)) for all a, b ≥ 0, we obtain n-[Φ d (d u )] ≤ Φ d (d u ) + + 6

  1] d is a standard Brownian sheet, kt := (kt 1 , . . . , kt d ) for k ∈ Z and the convergence happens in the Skhorohod space D([0, 1] d ) endowed with the uniform topology. Moreover, σ 2 = lim

		E[ S2 n,...,n ]
	n→∞	n d

Acknowledgment

The authors would like to thank in particular Dalibor Volný and Christophe Cuny for their helpful remarks during the writing of this article.