Magda Peligrad
email: peligrm@ucmail.uc.edu

Dalibor Volný
email: dalibor.volny@univ-rouen.fr

Quenched invariance principles for orthomartingale-like sequences

Keywords: random …elds, quenched central limit theorem, theorems started at a point, orthomartingales, coboundary. Mathematical Subject Classi…cation (2000): 60F05, 60G60, 60G42, 60G48

In this paper we study the central limit theorem and its functional form for random …elds which are not started from their equilibrium, but rather under the measure conditioned by the past sigma …eld. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random …elds by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clari…es the optimality of the moment conditions used for the validity of our results. Finally, by using the so called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random …elds.

Introduction and the quenched CLT

A very interesting type of convergence, with many practical applications, is the almost sure conditional central limit theorem and its functional form. This means that these theorems hold when the process is not started from its equilibrium but it is rather started from a …xed past trajectory. In the Markovian setting such a behavior is called a limit theorem started at a point. In general these results are known under the name of quenched limit theorems, as opposed to the annealed ones. A quenched CLT, for instance, is a stronger form of convergence in distribution and implies the usual CLT. There are examples in the literature showing that the annealed CLT does not necessarily implies the quenched one. See for instance [START_REF] Ouchti | A Conditional CLT which fails for ergodic components[END_REF] and [START_REF] Volný | An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process. Dependence in analysis, probability and number theory[END_REF].

The limit theorems started at a point or from a …xed past trajectory are often encountered in evolutions in random media and they are of considerable importance in statistical mechanics. They are also useful for analyzing Markov chain Monte Carlo algorithms.

In the context of random processes, this remarkable property is known for a martingale which is stationary and ergodic, as shown in Ch. 4 in [START_REF] Borodin | Limit theorems for functionals of random walks[END_REF] or on page 520 in [START_REF] Derriennic | The central limit thorem for Markov chains with normal transition operators started at a point[END_REF]. By using martingale approximations, this result was extended to larger classes of random variables by [START_REF] Cuny | Central limit theorem started at a point for stationary processes and additive functional of reversible Markov Chains[END_REF], [START_REF] Volný | Quenched central limit theorems for sums of stationary processes[END_REF], [START_REF] Cuny | On martingale approximations and the quenched weak invariance principle[END_REF], among others (for a survey see [START_REF] Peligrad | Quenched Invariance Principles via Martingale Approximation; in Asymptotic laws and methods in stochastics[END_REF].

A random …eld consists of multi-indexed random variables (X u ) u2Z d . An important class of random …elds are orthomartingales which have been introduced by [START_REF] Cairoli | Un theoreme de convergence pour martingales à indices multiples[END_REF] and further developed in Khoshnevisan (1982). They have resurfaced in many recent works. New versions of the central limit theorem for stationary orthomartingales can be found in [START_REF] Wang | A new criteria for the invariance principle for stationary random …elds[END_REF], Volny (2015Volny ( , 2019)), which complement the results in [START_REF] Basu | On functional central limit theorem for stationary martingale random …elds[END_REF], where a di¤erent de…nition of multiparameter martingale was used.

In order to exploit the richness of the martingale techniques several authors provided interesting su¢ cient conditions for orthomartingale approximations, such as [START_REF] Gordin | Martingale-coboundary representation for a class of stationary random …elds[END_REF], [START_REF] Volný | An invariance principle for stationary random …elds under Hannan's condition[END_REF], [START_REF] Cuny | A functional central limit theorem for …elds of commuting transformations via martingale approximation[END_REF], El Machkouri and Giraudo (2016), Peligrad and Zhang (2018 a), [START_REF] Giraudo | Invariance principle via orthomartingale approximation[END_REF], [START_REF] Volný | Martingale-coboundary representation for stationary random …eld[END_REF]. Other recent results involve random …elds which are functions of independent random variables as in El Machkouri et al. (2013) and [START_REF] Wang | A new criteria for the invariance principle for stationary random …elds[END_REF]. Peligrad and Zhang (2018 b) obtained necessary and suf-…cient conditions for an orthomartingale approximation in the mean square. These approximations make possible to obtain the central limit theorem (CLT) for a large class of random …elds. As in the case of a stochastic processes, a natural and important question is to get a quenched version of these CLT's. Motivated by this question, we obtain …rst a quenched CLT for orthomartingales. We show by examples that the situation is di¤erent for random …elds. An orthomartingale which satis…es the CLT may fail to satisfy the quenched CLT. The example we constructed also throws light on the optimality of the moment conditions we use in our main result. Finally, we extend the quenched CLT to its functional form and to a larger class of random …elds which can be decomposed into a orthomartingale and a coboundary. We shall apply our results to linear and nonlinear random …elds, often encounters in economics.

For the sake of clarity, due to the complicated notation, we shall explain in detail the case d = 2 and the proof of the quenched CLT. Then, in the subsequent sections, we shall discuss the general index set Z d and other extensions of these results.

Let ( ; K; P ) be a probability space, let T and S be two commuting, invertible, bimeasurable, measure preserving transformations from to ; and let F 0;0 be a sub-sigma …eld of K. For all (i; j) 2 Z 2 de…ne

F i;j = T i S j (F 0;0 ), i; j 2 Z: (1)
Assume the …ltration is increasing in i for every j …xed and increasing in j for every i …xed (i.e. F 0;0 F 0;1 and F 0;0 F 1;0 ). For all i and j we also de…ne the following sigma algebras generated by the unions of sigma algebras:

F i;1 = _ m2Z F i;m ; F 1;j = _ n2Z
F n;j and F 1;1 = _ n;m2Z F n;m : In addition assume the …ltration is commuting, in the sense that for any integrable variable X; with notation E a;b X = E(XjF a;b ); we have

E u;v E a;b X = E a^u;b^v X: (2)
We introduce the stationary sequence as following. De…ne a function X 0;0 : ! R; which is F 0;0 measurable, and the random …eld

X i;j (!) = X 0;0 (T i S j (!)): (3)
For the …ltration (F i;j ) de…ned by [START_REF] Argiris | Forcing divergence when the supremum is not integrable[END_REF] we call the random …eld (X i;j ) i;j2Z de…ned by (3) orthomartingale di¤erence …eld, if

E(X i;j jF u;v ) = 0 if either u < i or v < j: (4)
This de…nition implies that for any i …xed (X i;j ) j2Z is a sequence of martingale di¤erences with respect to the …ltration (F 1;j ) j2Z and also for any j …xed (X i;j ) i2Z is a sequence of martingale di¤erences with respect to the …ltration

(F i;1 ) i2Z : Set S n;v = X n 1 i=0 X v 1 j=0 X i;j : Below, ) denotes convergence in distribution.
The results in this paper are motivated by the following annealed CLT in [START_REF] Volný | A central limit theorem for …elds of martingale di¤erences[END_REF], which was extended to a functional CLT in [START_REF] Cuny | A functional central limit theorem for …elds of commuting transformations via martingale approximation[END_REF].

Theorem A Assume that (X i;j ) i;j2Z is de…ned by (3) and satis…es (4). Also assume that the …ltration (F i;j ) i;j2Z is de…ned by [START_REF] Argiris | Forcing divergence when the supremum is not integrable[END_REF] and satis…es [START_REF] Basu | On functional central limit theorem for stationary martingale random …elds[END_REF]. Assume that S (or T ) is ergodic and X 0;0 is square integrable, E(X 2 0;0 ) = 2 . Then, 1 (nv) 1=2 S n;v ) N (0; 2 ) when n ^v ! 1:

Let us point out that if S (or T ) is ergodic, then the Z 2 action generated by S and T is necessarily ergodic. However the ergodicity is not enough for Theorem A to hold. In Example 5.6. in [START_REF] Wang | A new criteria for the invariance principle for stationary random …elds[END_REF] and then in more detail by Volny (2015), a simple example of ergodic random …eld which does not satisfy the central limit theorem is analyzed. Starting with two sequences of i.i.d. random variables, centered with …nite second moments, (X n ) and (Y n ), the example is provided by the random …eld (Z i;j ); with Z i;j = X i Y j for all (i; j).

It should be noted that Theorem A has a di¤erent area of applications than Theorem 1 in [START_REF] Basu | On functional central limit theorem for stationary martingale random …elds[END_REF]. In this latter paper the …ltration is not supposed to be commuting. For a random …eld (X i;j ) i;j 1 their …ltration (K n;m ) is generated by the variables fX i;j : (j 1; 1 i n) [ (i 1; 1 j m)g: Suppose ( i;j ) are i.i.d., standard normal random variables. Then, Theorem A can be applied, for instance, to the random …eld (X i;j ); where X i;j = X 0;0 (T i S j (!)) with X 0;0 = 1;0 0;1 and F 0;0 = ( i;j ; i 0; j 0) but the result in [START_REF] Basu | On functional central limit theorem for stationary martingale random …elds[END_REF] cannot. On the other hand the random …eld (Y i;j ); de…ned by Y i;j = Y 0;0 (T i S j (!)) with Y 0;0 = P 1 k=1 a k ( k;0 + 0;k ) and P 1 k=1 ja k j < 1; can be treated by the result in [START_REF] Basu | On functional central limit theorem for stationary martingale random …elds[END_REF] but not by Theorem A.

It should also be noted that Theorem A allows to study the central limit theorem for orthomartingales which are not de…ned by a Bernoulli Z 2 -action.

The aim of this paper is to establish a quenched version of Theorem A. We denote by P ! ( ) = P ! 0;0 ( ) a version of the regular conditional probability P ( jF 0;0 )(!).

One of the results of this paper is the following theorem:

Theorem 1 Assume that the conditions of Theorem A are satis…ed. Then for P -almost all ! 2 ;

1 n S n;n ) N (0; 2 ) under P ! : (5)
In addition, if E(X 2 0;0 log(1 + jX 0;0 j)) < 1;

then for almost all all ! 2 ;

1 (nv) 1=2 S n;v ) N (0; 2 ) under P ! when n ^v ! 1: (7)
We would like to mention that, because by integration the quenched CLT implies the annealed CLT, the conclusion of Theorem 1 implies the CLT in Theorem A. However, when the summation on the rectangles is not restricted, the integrability assumption [START_REF] Billingsley | Probability and measures[END_REF] is stronger than in Theorem A. Let us also notice that the second part of Theorem 1 does not always hold under the assumption E(X 2 0;0 ) < 1. As a matter of fact we are going to provide an example to support this claim.

Theorem 2 Under the setting used in Theorem 1, there is a stationary sequence (X n;m ) n;m2Z satisfying (4), adapted to a commuting …ltration (F i;j ) i;j2Z ; with E(X 2 0;0 ln(1 + jX 0;0 j)) = 1; for any 0 < " < 1; E(X 2 0;0 ln 1 " (1 + jX 0;0 j)) < 1 and such that (S n;m = p nm) (n;m)2Z 2 does not satisfy the quenched CLT in [START_REF] Burkholder | Distribution function inequalities for martingales[END_REF].

We mention that, as a matter of fact, in our examples, both transformations constructed for the de…nition of (X n;m ) n;m2Z and for the …ltration (F i;j ) i;j2Z ; are ergodic. Also, this example satis…es the quenched CLT in [START_REF] Billingsley | Convergence of probability measures[END_REF].

The detailed proofs of these two theorems are contained in Section 2. Various extensions of Theorem 1 will be given in subsequent sections.

In Section 3 we formulate the functional form of the quenched CLT and we indicate how to prove it, by adapting the arguments from the proof of Theorem 1 and some other proofs of several known results.

For the sake of applications, in Section 4, we extend the results beyond orthomartingales, to a class of random …elds which can be decomposed into an orthomartingale and a generalized coboundary.

In Section 5 we show that Theorem 1 remains valid for random …elds indexed by Z d ; d > 2: The only di¤erence is that we replace condition (6) by E(X 2 0;0 log d 1 (1 + jX 0;0 j)) < 1: In Section 6 we apply our results to linear and nonlinear random …elds with independent innovations. Several useful results for our proofs are given in Section 7.

Proofs of Theorems 1 and Proof of Theorem 1

To …x the ideas, let us suppose that the transformation S is ergodic. Let us denote by T and Ŝ the operators on L 2 , de…ned by T f = f T and Ŝf = f S: Everywhere in the paper, for x real, we shall denote by [x] the integer part of x:

By using a truncation argument, we show …rst that, without restricting the generality, we can prove the theorem under the additional assumption that the variables are bounded. We shall introduce the following projection operators:

P i;j (X) = E i;j (X) E i;j 1 (X) E i 1;j (X) + E i 1;j 1 (X):
Let A be a positive integer. Denote X 0 i;j = X i;j I(jX i;j j A) and X " i;j = X i;j I(jX i;j j > A): Therefore, we can represent (X i;j ) as a sum of two orthomartingale di¤erences adapted to the same …ltration.

X i;j = P i;j (X 0 i;j ) + P i;j (X " i;j ):

Note that, jP 0;0 (X " 0;0 )j jX 0;0 j + E 1;0 jX 0;0 j + E 0; 1 jX 0;0 j + E 1;1 jX 0;0 j: Whence, by the properties of conditional expectation, E(X 0;0 ) 2 < 1 implies

E(P 0;0 (X " 0;0 )) 2 < 1 (9)
and E(X 2 0;0 log(1 + jX 0;0 j)) < 1 implies

E((P 0;0 (X " 0;0 )) 2 log(1 + j(P 0;0 (X " 0;0 ))j) < 1: (10) Set S 0 n;v = X n 1 i=0 X v 1 j=0 P i;j (X 0 i;j ) and S " n;v = X n 1 i=0 X v 1 j=0 P i;j (X " i;j ):
We shall show that, for almost all !;

lim A!1 lim sup n^v!1 P ! ( 1 (nv) 1=2 jS " n;v j > ") = 0:
By conditional Markov inequality, it is enough to show that

lim A!1 lim n^v!1 1 nv E 0;0 (S " n;v ) 2 = 0 a.s. ( 11
)
By the orthogonality of the orthomartingale di¤erences, we have that

1 nv E 0;0 ((S " n;v ) 2 ) = 1 nv X n 1 i=0 X v 1 j=0 E 0;0 (P i;j (X " i;j )) 2 : (12)
Note that the conditional expectation introduces a family of operators de…ned by

Q 1 (f ) = E 0;1 ( T f ) ; Q 2 (f ) = E 1;0 ( Ŝf ):
So, using (2), we can write

E 0;0 (P i;j (X " i;j )) 2 = Q i 1 Q j 2 (P 0;0 (X " 0;0 )) 2 :
Since Q 1 and Q 2 are integral preserving Dunford-Schwartz operators, by the ergodic theorem (see Theorem 3.5 in Ch. 6 in [START_REF] Krengel | Ergodic Theorems[END_REF], if we assume …nite second moment, by [START_REF] Cairoli | Un theoreme de convergence pour martingales à indices multiples[END_REF],

lim n!1 1 n 2 X n 1 i=0 X n 1 j=0 Q i 1 Q j 2 (P 0;0 (X " 0;0 )) 2 = E(P 0;0 (X " 0;0 )) 2 a.s.
If we assume E(X 2 0;0 log(1 + jX 0;0 j)) < 1 then, by [START_REF] Cuny | Central limit theorem started at a point for stationary processes and additive functional of reversible Markov Chains[END_REF] and Theorem 1.1 in Ch. 6, [START_REF] Krengel | Ergodic Theorems[END_REF], we obtain

lim n^v!1 1 nv X n 1 i=0 X v 1 j=0 Q i 1 Q j 2 (P 0;0 (X " 0;0 )) 2 = E(P 0;0 (X " 0;0 )) 2 a.s. ( 13
)
Clearly lim A!1 P 0;0 (X " 0;0 ) = 0 a.s. So, by the dominated convergence theorem,

lim A!1 E(P 0;0 (X " 0;0 )) 2 = 0;
and ( 11) is established. By Theorem 3.2 in [START_REF] Billingsley | Convergence of probability measures[END_REF], in order to establish conclusion [START_REF] Burkholder | Distribution function inequalities for martingales[END_REF] of Theorem 1, it is enough to show that for A …xed, for almost all ! 2 ;

1 (nv) 1=2 S 0 n;v ) N (0; 2 A ) under P ! as n ^v ! 1; and 2 A ! 2 as A ! 1: Above, 2 A = E(P 0;0 (X 0 0;0 )) 2 : Clearly, when A ! 1; 2 A ! 2 :
Therefore the result is established if we prove Theorem 1 for orthomartingale di¤erences which are additionally uniformly bounded.

So, in the rest of the proof, without restricting the generality, we shall assume that the variables (X i;j ) i;j2Z are bounded by a positive constant C.

Denote

F i;v = 1 v 1=2 X v 1 j=0 X i;j : (14)
We treat the double summation as a sum of a triangular array of martingale di¤erences (F i;v ) i 0 :

1 (nv) 1=2 S n;v = 1 n 1=2 X n 1 i=0 F i;v :
We shall apply Theorem 1 in [START_REF] Gänssler | Remarks on the functional central limit theorem for martingales[END_REF], given for convenience in Theorem 15 from Section 7, to

D n;i = F i;v = p n.
We have to show that for almost all !; both conditions of this theorem are satis…ed, namely we shall verify that lim

n!1 1 n E 0;0 j X [(n 1)t] i=0 (F 2 i;v 2 t)j = 0 a.s. ( 15
)
and

1 n E 0;0 max 0 i n 1 F 2 i;v is bounded a.s. ( 16
)
In order to check condition [START_REF] Eisner | Operator theoretic aspects of ergodic theory[END_REF], we use a blocking procedure. We verify it …rst with t = 1.

Let m 1 be a …xed integer and de…ne consecutive blocks of indexes of size m, I j (m) = f(j 1)m; :::; mj 1g: In the set of integers from 0 to n 1 we have u = u n (m) = [n=m] such blocks of integers and a last one containing less than m indexes. Practically, by the triangle inequality, we write

1 n j X n 1 i=0 (F 2 i;v 2 )j 1 n X u j=1 j X k2Ij (m) (F 2 k;v 2 )j + 1 n j X n 1 k=um (F 2 k;v 2 )j 1 u X u j=1 j 1 m X k2Ij (m) F 2 k;v 2 j + 1 n j X n 1 k=um (F 2 k;v 2 )j = I n;m + II n;m :
The task is now to show that

lim m!1 lim n^v!1 E 0;0 (I n;m ) = 0 a.s. ( 17
)
and lim

m!1 lim n^v!1 E 0;0 (II n;m ) = 0 a.s. ( 18
)
Let us treat …rst the limit of E 0;0 (I n;m ). Let N 0 be a …xed integer and consider n ^v > N 0 . By using the properties of the conditional expectations and (2) we obtain the following bound for E 0;0 (I n;m ) :

E 0;0 (I n;m ) = 1 u E 0;0 X u j=1 j 1 m X k2Ij (m) F 2 k;v 2 j = 1 u E 0;0 X u j=1 E (j 1)m;0 j 1 m X k2Ij (m) F 2 k;v 2 j = E 0;0 1 u X u 1 i=0 T im E 0;0 j 1 m X m 1 k=0 F 2 k;v 2 j E 0;0 1 u X u 1 i=0 T im (h m;N0 );
where we have used the notation

h m;N0 = sup v>N0 E 0;0 j 1 m X m 1 k=0 F 2 k;v 2 j:
Note that h m;N0 is bounded. Indeed, by the martingale property and the uniform boundedness of the variables by C, it follows that

h m;N0 2 + 1 m X m 1 k=0 sup v>N0 E 0;0 (F 2 k;v ) = 2 + 1 m X m 1 k=0 sup v>N0 E 0;0 ( 1 v X v 1 u=0 X 2 k;u ) 2 + C 2 :
By the ergodic theorem, (see Theorem 11.4

lim u!1 1 u X u 1 i=0 T im h m;N0 = E(h m; N 0 jI) = E I (h m; N 0 ) a.s.,
where I is the invariant sigma …eld for the operator T . Furthermore, we also have that 1 u

X u 1 i=0 T im h m;N0 2 + C 2 :
So, by Theorem 34.2 (v) in Billingsley (1995) (see Theorem 16 in Section 7) we derive that

lim u!1 E 0;0 1 u X u 1 i=0 T im h m;N0 = E 0;0 E I (h m; N 0 ) a.s.
Since the functions are bounded, by applying twice, consecutively, Theorem 16, we obtain that

lim N0!1 lim u!1 E 0;0 1 u X u 1 i=0 T im h m;N0 = E 0;0 E I ( lim N0!1 h m; N 0 ) a.s.
Clearly, because the variables are bounded, for every m …xed

E 0;0 E I ( lim N0!1 h m; N 0 ) = E 0;0 E I (lim sup v E 0;0 j 1 m X m 1 k=0 F 2 k;v 2 j) E 0;0 E I E 0;0 (lim sup v E 1;0 j 1 m X m 1 k=0 F 2 k;v 2 j):
Now, by using again the fact that the variables are bounded and using Theorem 16, in order to show that

lim m!1 E 0;0 E I ( lim N0!1 h m; N 0 ) = 0 P -a.s. it is enough to show that lim m!1 lim sup v E 1;0 j 1 m X m 1 k=0 F 2 k;v 2 j = 0 a.s. ( 19
)
With this aim, we note …rst that by the ergodicity of S and the fact that the variables are bounded, it follows that, for any k;

lim v!1 E 1;0 F 2 k;v = lim v!1 1 v E 1;0 ( X v 1 j=0 X 2 k;j ) = 2 : (20)
Denote P ! 1;0 ( ) = P ( jF 1;0 ): We also know that for any k, by the quenched CLT for stationary martingale di¤erences (see, for instance, Ch. 4 in [START_REF] Borodin | Limit theorems for functionals of random walks[END_REF] or Derrienic and Lin (2001)), for almost all !; F k;v ) N k under P ! 1;0 , where N k is a centered normal random variable with variance 2 : Therefore, by the su¢ ciency part of the convergence of moments associated to weak convergence, namely Theorem 3.6 in Billingsley (1999), we have that where (N 0 ; N 1 ; :::; N m 1 ) is a Gaussian vector of centered normal variables with variance 2 . But since (F j;v ) j2Z are uncorrelated it follows by ( 21) that the variables in (N i ) i 0 are also uncorrelated and therefore (N i ) i 0 is an i.i.d. sequence. By the continuous mapping theorem,

(F 2 k;v ) v
1 m X m 1 k=0 (F 2 k;v 2 ) ) 1 m X m 1 k=0 (N 2 k
2 ) under P ! 1;0 for almost all !.

By [START_REF] Giraudo | Invariance principle via orthomartingale approximation[END_REF] it follows that (

P m 1 k=0 (F 2 k;v 2
)) v 1 is also uniformly integrable, so we can apply the convergence of moments from Theorem 3.5 in Billingsley (1999). Therefore, denoting by E the expectation in rapport with the probability on the space where the variables (N k ) 0 s are de…ned, we obtain

lim v!1 E 1;0 j 1 m X m 1 k=0 (F 2 k;v 2 )j = Ej 1 m X m 1 k=0 (N 2 k
2 )j a.s.

By letting m ! 1 and using the law of large numbers for an i.i.d. sequence, we obtain lim

m!1 E(j 1 m X m 1 k=0 (N 2 k
2 )j = 0: Therefore ( 19) follows. As a consequence, we obtain [START_REF] El Machkouri | Orthomartingale-coboundary decomposition for stationary random …elds[END_REF].

In order to treat the term (18), we estimate

E 0;0 (II n;m ) = E 0;0 1 n j X n 1 k=um (F 2 k;v 2 )j m n 2 + E 0;0 1 n X n 1 k=um F 2 k;v m n 2 + 1 n X n 1 k=um 1 v X v 1 j=0 E 0;0 X 2 k;j m n ( 2 + C 2 ) a.s.
Whence, [START_REF] Gänssler | Remarks on the functional central limit theorem for martingales[END_REF] follows, by passing to the limit …rst with n ! 1 followed by m ! 1.

Overall, we have shown that

lim n^v!1 1 n E 0;0 j X n 1 u=0 (F 2 u;v
2 )j = 0 a.s.

If we replace now n 1 by [(n 1)t] we easily see that we also have convergence to t 2 and (15) follows: It remains to verify the second condition of Theorem 15, namely to prove [START_REF] El Machkouri | A central limit theorem for stationary random …elds[END_REF]. To show it, note that, by the martingale property,

1 n E 0;0 ( max 0 i n 1 F 2 i;v ) 1 n E 0;0 ( X n 1 i=0 F 2 i;v ) = 1 nv ( X n 1 i=0 X v 1 u=0 E 0;0 (X 2 i;u )) C 2 a.s.
The proof of the theorem is now complete.

Proof of Theorem 2

We start with an i.i.d. random …eld ( n;m ) n;m2Z de…ned on a probability space ( ; K; P ) with the distribution

P ( 0;0 = 1) = P ( 0;0 = 1) = 1=2: (22)
Without restricting the generality we shall de…ne ( u ) u2Z 2 in a canonical way on the probability space = R Z 2 , endowed with the …eld B; generated by cylinders. Then, if ! = (x v ) v2Z 2 , we de…ne 0 u (!) = x u . We construct a probability measure P 0 on B such that for all B 2 B, any m and u 1 ; :::; u m we have P 0 ((x u1 ; :::; x um ) 2 B) = P (( u1 ; :::; um ) 2 B):

The new sequence ( 0 u ) u2Z 2 is distributed as ( u ) u2Z 2 and re-denoted by ( u ) u2Z 2 . We shall also re-denote P 0 as P: Now on R Z 2 we introduce the operators

T u ((x v ) v2Z 2 ) = (x v+u ) v2Z 2 :
Two of them will play an important role, namely when u =(1; 0) and when u =(0; 1): By interpreting the indexes as notations for the lines and columns of a matrix, we shall call

T ((x u;v ) (u;v)2Z 2 ) = (x u+1;v ) (u;v)2Z 2
the vertical shift and

S((x u;v ) (u;v)2Z 2 ) = (x u;v+1 ) (u;v)2Z 2
the horizontal shift. Introduce the …ltration F n;m = ( i;j ; i n; j m) and notice that this …ltration is commuting. We assume K = F 1;1 : The transformations T and S are invertible, measure preserving, commuting and ergodic. Furthermore T i;j = T i S j : For a measurable function f de…ned on R Z 2 de…ne

X j;k = f (T j S k ( a;b ) a 0;b 0 ): (23)
We notice that the variables are adapted to the …ltration (F n;m ) n;m2Z .

As an important step for constructing our example we shall establish the following lemma:

Lemma 3 For every n and every " > 0 we can …nd a set F = F (n; ") which is F 0;0 measurable and such that

P (F ) 1 n 2 (1 "):
Furthermore, for any 0 i; j n 1; 0 k; ` n 1 with (i; j) 6 = (k; `) we have

P (T 1 i;j F \ T 1 k;l F ) = 0: (24)
Proof of Lemma 3.

Let n be an integer and let " > 0: By using Rokhlin lemma (see Theorem 17 in Section 7), construct B 2 K with

P (B) (1 " 2 ) 1 n 2 (25)
and for 0 i; j n 1, T 1 i;j B are disjoint for distinct pair of indexes. Since K is generated by the …eld [ n F n ; we can …nd a set E in [ n F n such that

P (B E) < " 8n 4 : (26)
Since E belongs to

[ n F n ; there is a m such that E 2 F m : So T m (E) 2 F 0 : Denote G = T m (E) and set F = G n [ (i;j)2D T 1 i;j G;
where D = f0 i; j n 1; (i; j) 6 = (0; 0)g. Note now that for all (i; j) 2 D;

P (F \ T 1 i;j F ) = 0;
which implies [START_REF] Krengel | Ergodic Theorems[END_REF]. Also, by stationarity,

P (F ) = P (E) P (E \ ([ (i;j)2D T 1 i;j E)) P (E) X (i;j)2D P (E \ T 1 i;j E):
But for (i; j) 2 D;

P (E \ T 1 i;j E) 2P (E n B) " 4n 4 :
Therefore, by the above considerations, ( 26) and ( 25) we obtain

P (F ) P (E) " 4n 2 P (B) " 8n 4 " 4n 2
1 " n 2 :

Next, we obtain a lemma which is the main step in the construction of the example. In the sequel, we use the notation a n b n for lim n!1 a n =b n = 1:

Lemma 4 There is a strictly stationary random …eld of integrable positive random variables (U i;j ) i;j2Z ; coordinatewise ergodic, such that for any 0 < " < 1; EjU 0;0 j ln 1 " (1 + jU 0;0 j) < 1 and such that for almost all !; (U n;v =nv) n;v2Z is not tight under P ! : Proof of Lemma 4.

By Lemma 3, for n

2 and " = 1=2; we can …nd sets F n 2 F n; n such that P (F n ) = 1=2n 2 and such that for any 0 i; j n 1; 0 k; ` n 1 with (i; j) 6 = (k; `) we have P (T 1 i;j F n \ T 1 k;l F n ) = 0: Now, we consider independent copies of the probability space ( ; K; P ); denoted by ( (m) ; K (m) ; P (m) ) m 1 ; and introduce the product space

= Q 1 m=1 (m)
endowed with the sigma algebra generated by cylinders, K = Q 1 m=1 K (m) . We also introduce on K the product probability P = Q 1 m=1 P (m) ; P (m) = P . In this space consider sets F (n) n which are products of with the exception of the n-th coordinate which is F n :

On ; de…ne a random variable f n by the following formula:

f n = n ln 2 n 1 F (n) n : (27)
Let A n be the following event:

A n = fthere are i; j, ln n i; j n 1; such that f n T i;j =ij 1g:

where T i;j = (T i;j ; T i;j ; :::

): Since f n T i;j is Q 1 m=1 F (m)
0;0 measurable, for ! 2 A n ; there are i; j, ln n i; j n 1; such that

P ! (f n T i;j =ij 1) = 1: (28)
Note now that f n T i;j =ij 1 if and only if 1

F (n) n T i;j ij(ln n) 2 =n; if and only if ! 2 (T i;j ) 1 (F (n) n ) and ij n=(ln n) 2 .
Then, the probability of A n can be computed as:

P(A n ) = P( [ D T 1 i;j (F (n) n ) = P ( [ D T 1 i;j (F n ));
where the union and have indexes in the set D = fij (n 1)=(ln n) 2 ; ln n i; j n 1g: By Lemma 3, it follows that

P(A n ) = P (F n ) X ln n j n 1 X ln n i (n 1)=j(ln n) 2 1 n ln n 2n 2 (ln n) 2 = 1 2n ln n : Therefore X n 2 P(A n ) = X n 2 1 2n ln n = 1:
By the second Borel-Cantelli lemma, P(A n i.o.) = 1. This means that almost all ! 2 belong to an in…nite number of A n . Whence, taking into account [START_REF] Peligrad | Quenched Invariance Principles via Martingale Approximation; in Asymptotic laws and methods in stochastics[END_REF], for almost all ! 2 and every positive B;

lim sup i^j!1 P ! (f m T i;j =ij B) = 1: (29) De…ne now U 0;0 = X n 2 f n and U i;j = X n 2 f n T i;j : (30)
Let us estimate the Luxembourg norm of U 0;0 in the Orlicz space generated by the convex function g(x) = x ln 1 " (1 + x) for x > 0, 0 < " < 1. For each n 2 N

jjf n jj g = inf f : E( f n ln 1 " (1 + f n )) 1g :
By the de…nition of f n , we have

E( f n ln 1 " (1 + f n )) = P (F n ) n ln 2 n ln 1 " (1 + n ln 2 n ) = 1 2 n ln 2 n ln 1 " (1 + n ln 2 n ):
From this identity we see that, after some computations, that for n su¢ ciently large jjf n jj g 1 n ln 1+"=2 n :

Clearly, we have jjU 0;0 jj g X n 2

jjf n jj g < 1:

It remains to note that, by de…nition (30), U i;j f n T i;j . Therefore, by [START_REF] Peligrad | On the normal approximation for random …elds via martingale methods[END_REF] we also have for almost all ! 2 and every positive B;

lim sup i^j!1 P ! (U i;j =ij B) = 1
and the conclusion of this lemma follows by letting B ! 1.

End of proof of Theorem 2

On the space constructed in Lemma 4 de…ne the independent random variables 0 i;j (! 1 ; ! 2 ; :::) = i;j (! 1 ) and the random variables X i;j = 0 i;j U 1=2 i 1;j 1 ; where (U i;j ) i;j2Z and ( i;j ) i;j2Z are as in Lemma 4. Note that (X i;j ) i;j2Z is a sequence of orthomartingale di¤erences with respect to

Q 1 m=1 F (m) i;j , where
F (m)
i;j are independent copies of F i;j . According to Lemma 4 for almost all ! 2 we have lim

B!1 lim sup i^j!1 P ! (jX i;j j= p ij B) = 1:
If we assume now that (S n;m = p nm) n;m 1 satis…es the quenched limit theorem (or it is "quenched" tight), because U 1=2 i 1;j 1 = jX i;j j jS i;j j + jS i 1;j j + jS i;j 1 j + jS i 1;j 1 j; then necessarily the …eld (jX m;m j= p nm) n;m 1 should be tight under P ! ; for almost all !, which leads to a contradiction. Note that, by [START_REF] Volný | A central limit theorem for …elds of martingale di¤erences[END_REF], for any 0 < " < 1 we have EX 2 0;0 ln 1 " (1 + jX 0;0 j) < 1: For this example EX 2 0;0 ln(1 + jX 0;0 j) = 1; since otherwise the quenched result follows by Theorem 1.

Quenched functional CLT

In this section we formulate the functional CLT, which holds under the same conditions as in Theorem 1. For (s; t) 2 [0; 1] 2 ; we introduce the stochastic process

W n;v (t; s) = 1 p nv S [nt];[vs] :
We shall establish the following result. Denote by (W (t; s)) (t;s)2[0;1] 2 the standard 2-dimensional Brownian sheet.

Theorem 5 Under the setting of Theorem 1, if we assume that E(X 2 0;0 ) < 1 then, for P -almost all !; the sequence of processes (W n;n (t; s)) n 1 converges in distribution in D([0; 1] 2 ) endowed with the uniform topology to W (t; s); under P ! . If we assume now that (6) holds, then for P -almost all !; the sequence (W n;v (t; s)) n;v 1 converges in distribution to W (t; s); as n ^v ! 1 under P ! .

Proof of Theorem 5

Let us …rst prove the second case, when n ^v ! 1: As usual, the proof of this theorem involves two steps, namely the proof of the convergence of the …nite dimensional distributions to the corresponding ones of the standard 2dimensional Brownian sheet and tightness.

The proof of the convergence of …nite dimensional distribution follows, up to a point, the proof of the corresponding result in Cuny et al. ( 2015), which will be combined with the method of proof in Theorem 1. As explained in Subsection 3.2 in [START_REF] Cuny | A functional central limit theorem for …elds of commuting transformations via martingale approximation[END_REF], in order to establish the convergence of the …nite dimensional distributions, we have to show that for almost all ! 2 ; and for any partitions 0 t 1 ::: t K 1 and 0 s 1 ::: s K 1; we have

1 p nv X K k=1 X K `=1 a k;`X [nt k ] 1 i=[nt k 1 ] X [vs `] 1 j=[vs ` 1 ] X i;j ) N (0; ) under P ! ; (32)
where

= 2 P K k=1 P K `=1 a 2 k;`( t k t k 1 )(s ` s ` 1 ):
In order to establish this weak convergence we follow step by step the proof of Theorem 1. We shall just mention the di¤erences. The …rst step is to decompose X i;j as in formula [START_REF] Borodin | Limit theorems for functionals of random walks[END_REF] and to show the negligibility of the term containing X " i;j : This is the only step where we need di¤erent moment conditions according to whether indexes in the sum are restricted or not. By using simple algebraic manipulations, the triangle inequality along with Theorem 3.2 in Billingsley (1999), we can easily see that this term is negligible P -a.s. for the convergence in D([0; 1] 2 ) endowed with the uniform topology, if, for every " > 0 lim A!1 lim sup n^v!1 P 0;0 ( max

1 i n max 1 j v j X i k=1 X j `=1 P k;`( X " k;`) j > " p nv) = 0 a.s.
But by using Cairoli's maximal inequality for orthomartinagles (see Theorem 2.3.1 in Khoshnevisan, 2002, p. 19) the proof is reduced to showing [START_REF] Cuny | On martingale approximations and the quenched weak invariance principle[END_REF], which was already proved in Theorem 1. Without loss of generality we redenote P i;j (X 0 i;j ) by X i;j and assume it is bounded by a positive constant C. We continue the steps of the proof in Theorem 1 with the exception that we replace F i;v in de…nition [START_REF] Derriennic | The central limit thorem for Markov chains with normal transition operators started at a point[END_REF] by

F k;i;v = 1 p v X K `=1 a k;`X [vs `] 1 j=[vs ` 1 ] X i;j ;
where

[nt k 1 ] i [nt k ] 1; 1 k K: We also replace 2 by 2 k = 2 P K `=1 a 2 k;`( s ` s ` 1 ) and h m;N0 by h k;m;N0 = sup v>N0 E 0;0 j 1 m X m 1 i=0 F 2 k;i;v 2 k j:
For instance, let us convince ourselves that (20) holds. Indeed by the ergodicity of S and the fact that the variables are bounded

lim v!1 E 1;0 F 2 k;i;v = lim v!1 1 v E 1;0 ( X K `=1 a k;`X [vs `] 1 j=[vs ` 1 ] X 2 i;j ) = 2 k :
After we verify the conditions of Theorem 15 for the triangular array of martingale di¤erences

(F k;i;v ) [nt k 1 ] i [nt k ] 1; 1 k K ,
we obtain the result in (32) by applying the CLT in Theorem 15.

For proving tightness we shall verify the moment condition given in relation (3) in [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF] and then the tightness follows from Theorem 3 in the same paper. To verify it is enough to compute the 4 th moment of an increment of the process W n;v (t; s) on the rectangle A = [t 1 ; s 1 ) [t 2 ; s 2 ): That is E( 4 (A)) where

(A) = 1 p nv X [nt2] 1 i=[nt1] X [vs2] 1 j=[vs1]
X i;j :

By applying Burkholder's inequality twice consecutively, and taking into account that the variables are bounded by C; for a positive constant K we obtain

E ! ( 4 (A)) KC 4 (t 2 s 1 ) 2 (t 2 s 2 ) 2 = KC 4 2 (A);
It should be noted that [START_REF] Giraudo | Invariance principle via orthomartingale approximation[END_REF] have shown that if

sup u;v 0 E((E 0;0 (S n;v )) 2 ) < 1; (37)
then the decomposition (34) holds and all the variables are in L 2 : As a matter of fact this is also a necessary condition for (34). The only condition speci…c to L 2 needed for his proof is the re ‡exivity of L 2 : Since the Orlicz space L ' generated by the function

'(x) = x 2 log(1 + x) : [0; 1) ! [0; 1)
is re ‡exive (see Theorem 8 in Milnes (1957)), the proof of Theorem 2.1 in Giraudo is also valid in this context. It follows that if

sup u;v 0 E('(jE 00 (S n;v )j)) < 1; (38)
then the decomposition in (34) holds all the functions are in L ' : The reciprocal is also true. As a matter of fact, by combining Theorem 7 with this result we deduce the following corollary:

Corollary 8 Let us assume that the random …eld (X i;j ) i;j2Z , de…ned by (3), adapted to the commuting …ltration (F i;j ) i;j2Z ; de…ned by [START_REF] Argiris | Forcing divergence when the supremum is not integrable[END_REF], satis…es [START_REF] Wang | A new criteria for the invariance principle for stationary random …elds[END_REF].

Then lim n^v!1 (nv) 1 E(S 2 n;v ) = c 2 .
If in addition we assume that S (or T ) is ergodic, then for almost all ! 2 ; (35) holds. Also, if condition (38) is satis…ed, then for almost all ! 2 ; (36) hods.

Proof of Theorem 7

Consider …rst that the indexes n and m are varying independently. Denote by m i;j = m 0;0 T i;j and M k;`= P k 1 i=0 P ` 1 j=0 m i;j : We shall establish [START_REF] Volný | Martingale-coboundary representation for stationary random …eld[END_REF]. A simple computation shows that (S k;` M k;`) = p nv is the sum of the following three terms:

1 p nv k 1 X i=0 ` 1 X j=0 T i Ŝj (I T )m 0 0;0 = 1 p nv ` 1 X j=0 Ŝj (m 0 0;0 T k m 0 0;0 ) = R 1 (k; `); 1 p nv k 1 X i=0 ` 1 X j=0 T i Ŝj (I Ŝ)m " 0;0 = 1 p nv k 1 X i=0 T i (m " 0;0 Ŝ`m" 0;0 ) = R 2 (k; `); 1 p nv k 1 X i=0 ` 1 X j=0 T i Ŝj (I T )(I Ŝ)Y 0;0 = 1 p nv (I Ŝ`) (I T k )Y 0;0 = R 3 (k; `):
In order to treat the last term, note that max

1 k n;1 ` v jR 3 (k; `)j 4 p nv max 0 i n max 0 j v jY i;j j:
Let A be a positive integer. By truncation at the level A we obtain the following bound

1 nv max 0 i n max 0 j v jY i;j j 2 A 2 nv + 1 nv n X i=0 v X j=0 Y 2
i;j I(jY i;j j > A):

Because of the stationarity and the fact that in the second part of Theorem 7 we imposed condition [START_REF] Billingsley | Probability and measures[END_REF], by the ergodic theorem for stationary random …elds (see Theorem 1.1 in Ch.6, [START_REF] Krengel | Ergodic Theorems[END_REF]) it follows that for every A;

lim n^v!1 1 nv n X i=0 v X j=0 Y 2
i;j I(jY i;j j > A) = E(Y 2 0;0 I(jY 0;0 j > A)):

Therefore lim A!1 lim n^v!1 jR 3 (n; v)j = 0 P a.s. By Fubini's theorem, it follows that the limit is 0 also under P ! , for almost all !.

The terms R 1 (k; `) and R 2 (k; `) are treated similarly, with small di¤erences. Let us treat the …rst one only. It is convenient to truncate at a positive number A. Let m 0 j;k = m 0 j;k I(jm 0 j;k j A) E j;k 1 m 0 j;k I(jm 0 j;k j A)+ m 0 j;k I(jm 0 j;k j > A) E j;k 1 m 0 j;k I(jm 0 j;k j > A):

We shall use the following bound:

E 0;0 max 1 k n;1 ` v R 2 1 (k; `) 2E 0;0 max 1 k n;1 ` v ( ` 1 X j=0 m 0 j;k ) 2 8A 2 v + 2E 0;0 max 1 k n;1 ` v ( ` 1 X j=0 m 0 j;k I(jm 0 j;k j > A) E j;k 1 m 0 j;k I(jm 0 j;k j > A)) 2 8A 2 v + 2 n X k=1 E 0;0 max 1 ` v ( ` 1 X j=0 m 0 j;k I(jm 0 j;k j > A) E j;k 1 m 0 j;k I(jm 0 j;k j > A)) 2 :
Now, by the Doob's maximal inequality

1 nv E 0;0 max 1 k n;1 ` v R 2 1 (k; `) 8A 2 n + 2 nv n X k=1 v 1 X j=0 E 0;0 (m 0 j;k I(jm 0 j;k j > A) E j;k 1 m 0 j;k I(jm 0 j;k j > A)) 2 8A 2 n + 4 nv n X k=1 v 1 X j=0 E 0;0 (m 0 j;k I(jm 0 j;k j > A)) 2 = 8A 2 n + 4 nv n X k=1 v 1 X j=0 Q j 1 Q k 2 [(m 0 0;0 ) 2 I(jm 0 0;0 j > A)]:
We let n ^v ! 1 and we use Theorem 1.1 in Ch. 6 of [START_REF] Krengel | Ergodic Theorems[END_REF]. It follows that, for every A

lim n^v!1 1 nv E 0;0 max 1 k n;1 ` v R 2 1 (k; `) = E(m 0 0;0 ) 2 I(jm 0 0;0 j > A):
Then, we let A ! 1: This completes the proof of [START_REF] Volný | Martingale-coboundary representation for stationary random …eld[END_REF]. The result follows by using the second part of Theorem 5 along with Theorem 3.2 in [START_REF] Billingsley | Convergence of probability measures[END_REF]. Now for the situation n = m ! 1; the proof is similar with the di¤erence that we use Theorem 3.5 in Ch. 6 in [START_REF] Krengel | Ergodic Theorems[END_REF] instead of Theorem 1.1 in the same chapter together with the …rst part of Theorem 5.

Remark 9 If we take Y 0;0 , in the martingale-coboundary decomposition (34), to be the function U 1=2 0;0 found in the proof of Lemma 4, then for almost all !,

R 3 (n; v) = 1 p nv n 1 X i=0 v 1 X j=0 T i Ŝj (I T )(I Ŝ)Y 0;0
does not converge to 0 in probability P ! when n ^v ! 1: Therefore if only the existence of the second moment is assumed or even if EY 2 0;0 ln 1 " (1+jY 0;0 j) < 1 for some 0 < " < 1; this coboundary could spoil the quenched weak convergence. This is in sharp contrast with the dimension 1. Recall that in dimension 1, when we have a martingale-coboundary decomposition X 0 = D 0 +G 0 T G 0 with D 0 a martingale di¤ erence and G 0 2 L 2 ; then the coboundary G 0 T G 0 does not spoil the quenched invariance principle (see Theorem 8.1 in [START_REF] Borodin | Limit theorems for functionals of random walks[END_REF], which is due to Gordin and Lifshits). In higher dimension, in general, we need stronger moment conditions not only for martingale di¤ erences but also for the cobounding function Y 0;0 .

The case of d-indexed random …eld

In this section we formulate our results and indicate their proofs for random …elds indexed by Z d with d > 2: The proofs are based on induction arguments. When we add on unrestricted d-dimensional rectangles the moment conditions will depend on d. By u =(u 1 ; u 2 ; :::; u d ) we denote elements of Z d . Let us suppose that T = (T i ) 1 i d are d commuting, invertible, measure preserving transformations from to and let F 0 be a sub-sigma …eld of K. For all u 2 Z d de…ne F u = T u (F 0 ); where T u is the following composition of operators:

T u = Q n i=1 T ui i
: Assume the …ltration is coordinatewise increasing and commuting, in the sense that for any integrable variable we have E u E a X = E a^u X; where a ^u means coordinatewice minimum and we used the notation E u X = E(XjF u ). We introduce the stationary …eld by starting with a F 0 measurable function X 0 : ! R and then de…ne the random …eld

X k (!) = X 0 (T k (!)) = X 0 (T k1 1 ::: T k d d ):
The operator T is de…ned on L 2

and therefore which is bounded under condition (39). Indeed, condition (39) implies that sup n 1 sup i 0 jb n;i j < 1; and then, after simple algebraic manipulations we can …nd a positive constant K such that E('(jb n;i 0 j)) Kb 2 n;i (E('(j 0 j)) + E( 2 0 )):

E(E 2 (S n jF 0 )) = X i 0 ( X 0 k n 1 a k+i ) 2 E( 2
It remains to apply the second part from Corollary 8 and Remark 11 in order to obtain the second part of the example.

Another class of nonlinear random …elds are the Volterra processes, which play an important role in the nonlinear system theory.

Example 13 Let ( n ) n2Z d be a random …eld of independent random variables, identically distributed, centered and with …nite second moment. De…ne take as innovations the random …eld ( n;m ) n:m2Z having as columns independent copies of a stationary and ergodic martingale di¤ erences sequence. In this case the …ltration generated ( n;m ) n;m2Z is also commuting. As a matter of fact a commuting …ltration could be generated by a stationary random …eld ( n;m ) n;m2Z where the columns are independent, i.e. m = ( n;m ) n2Z are independent.

Auxiliary results

The following is Theorem 1 in Gänssler and Häusler (1979) (see also [START_REF] Gänssler | Dependence in Probability and Statistics[END_REF], pages 315-317).

Theorem 15 Assume that (D n;k ) 1 k n is a triangular array of martingale differences adapted to an increasing …ltration (F n;k ) k . Assume We mention now Theorem 34.2 (v) in [START_REF] Billingsley | Probability and measures[END_REF]. Further reaching results including comments of the sharpness of the result below can be found in [START_REF] Argiris | Forcing divergence when the supremum is not integrable[END_REF].

Theorem 16 Assume that the sequence of random variables (X n ) n 0 converges a.s. to X and there is an integrable and positive random variable Y such that jX n j Y a.s. for all n 0. Let F be a sigma algebra. Then the sequence (E(X n jF)) n 0 converges a.s. to E(XjF):

The following is a particular case of Theorem 2 in Ornstein and Weiss (1980) known under the name of Rokhlin lemma for amenable actions. Theorem 17 Let ( ; K; P ) be a nonatomic probability space and let T be a measure preserving action of Z 2 into T : Z 2 ! that is ergodic. Then, for all " > 0 and n 2 N , there is a set B = B(n; ") 2 K such that for 0 i; j n 1, T 1 i;j B are disjoint for distinct indexes (i; j) and P (B) 1 n 2 (1 "):

  The result follows for S n by applying the …rst part of Corollary 8.On the other hand, by the Rosenthal inequality for independent random variables (see relation 21.5 in Burkholder (1973)), applied with the function '(x) = x 2 log d 1 (1 + jxj); there is a positive constant C such that E('(jE(S n jF 0 )j)) C' jb n;i 0 j));

0 k j 1 a

 1 X k = X (u;v) (0;0) a u;v k u k v ;where a u;v are real coe¢ cients with a u;u = 0 andP u;v 0 a 2 u;v < 1. Denote c u;v (j) = X k+u;k+v and d 2 u;v (j) = c 2 u;v (j) + c 2 v;u (j):Then the quenched functional CLT in (40) holds. If in addition we assume that E( 2 0 log d 1 (1 + j 0 j)) < 1, then the quenched functional CLT in (41) holds for sums of variables in a general d-dimensional rectangle.Proof of Example 13.

  ! P 2 t and max 1 k n jD n;k j is uniformly integrable. Then S [nt] ) W (t); where S [nt] = P [nt] k=1 D n;k and W (t) is a standard Brownian measure. In particular S n ) N (0; 2 ).

Acknowledgements. This research was supported in part by the NSF grants DMS-1512936 and DMS-1811373. The …rst author would like to o¤er special thanks to the University of Rouen Normandie, where a large portion of this research was accomplished during her visit of the Department of Mathematics. The authors wish to thank the referee for many in-depth comments, suggestions, and corrections, which have greatly improved the manuscript.

where is the Lebesgue measure on [0; 1] 2 : If B is a neighboring rectangle of A, by the Cauchy-Schwatz inequality we have

Therefore the moment condition in relation (3) in [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF] is veri…ed with = 4 and = 2:

4 Quenched functional CLT via coboundary decomposition Now we indicate a larger class than the orthomartingale, which satis…es a quenched functional CLT. A fruitful approach is to approximate S m;n by an orthomartingale M n;m in a norm that makes possible to transport the quenched functional CLT given in Theorem 5. Such an approximation is of the form: for every " > 0; lim sup n^v!1 P ! ( max

The random …elds we consider can be decomposed into an orthomartingale and a generalized coboundary and therefore satisfy [START_REF] Volný | Martingale-coboundary representation for stationary random …eld[END_REF]. This type of orthomartingale approximation, so called martingale-coboundary decomposition, was introduced for random …elds by [START_REF] Gordin | Martingale-coboundary representation for a class of stationary random …elds[END_REF] and studied by El Machkouri and Giraudo (2016), [START_REF] Giraudo | Invariance principle via orthomartingale approximation[END_REF] and [START_REF] Volný | Martingale-coboundary representation for stationary random …eld[END_REF].

De…nition 6

We say that a random …eld (X i;j ) i;j2Z , de…ned by (3), adapted to the commuting …ltration (F i;j ) i;j2Z ; de…ned by [START_REF] Argiris | Forcing divergence when the supremum is not integrable[END_REF], admits a martingalecoboundary decomposition if

with m 0;0 an orthomartingale di¤ erence (satisfying (4)), m 0 0;0 a martingale difference in the second coordinate and m " 0;0 a martingale di¤ erence in the …rst coordinate. All these functions are F 0;0 measurable.

We shall obtain the following generalization of Theorem 5:

Theorem 7 Let us assume that the decomposition (34) holds with all the variables square integrable and S (or T ) is ergodic. Then for almost all ! 2 ;

where (W (t; s)) (t;s)2[0;1] 2 is the standard 2-dimensional Brownian sheet and c 2 = E(m 2 0;0 ). If we assume that all the variables involved in the decomposition (34) satisfy (6) then, for almost all ! 2 ;

as T(f ) =f T: For the …ltration (F u ) u2Z d ; de…ned as above, we call the random …eld (X u ) u2Z d orthomartingale di¤erence if E(X u jF i ) = 0 when at least one coordinate of i is strictly smaller that the corresponding coordinate of u. We also use the notation i u; where the inequality is coordinatewise and jnj = n 1 ::: n d : Finally denote S n = X 0 i n 1

X i : In this context we have:

Theorem 10 Assume that there is an integer i,

Then, for almost all ! 2 ; 1 n d=2 S (n;n;:::;n) ) W (t 1 ; :::; t d ) under P ! when n ! 1 :

then for almost all ! 2 ; 1 jnj 1=2 S (n1;n2;:::;n d ) ) W (t 1 ; :::; t d ) under P ! when min

Remark 11 Both Theorems 5 and 7 also hold for the multi-indexed random …eld (X u ) u2Z d de…ned above.

We shall indicate how to prove these results by induction. We shall follow step by step the proof of Theorem 1 with the following di¤erences. Without restricting the generality, let us assume that the operator T i is ergodic for an i, 2 i d: We de…ne now the d-dimensional projection operators. By using the commutative property of the …ltrations it is convenient to de…ne: P u (X) = P u1 P u2 ::: P u d (X);

where

Above we used the notation u j for a vector which has the same coordinates as u with the exception of the j-th coordinate, which is u j 1. For instance when d = 3; P u2 (Y ) = E(Y jF u1;u2;u3 ) E(Y jF u1;u2 1;u3 ): We can easily see that, by using the commutativity property of the …ltration, this de…nition is a generalization of the case d = 2. We note that, by using this de…nition of P u (X); the truncation argument in Theorem 1 remains unchanged if we replace the index set Z 2 with Z d : We point out the following two di¤erences in the proof of Theorem 10. One di¤erence is that, for the validity of the limit in (13) when min 1 i d n i ! 1; in order to apply the ergodic theorem for Dunford-Schwartz operators, conform to Ch. 6 Theorem 2.8 and Theorem 1.1 in [START_REF] Krengel | Ergodic Theorems[END_REF], we have to assume that E[X 2 0;0 log d 1 (1 + jX 0;0 j)] < 1. After we reduce the problem to the case of bounded random variables, we proceed with the proof of the CLT by induction. More precisely, we write the sum in the form The tightness in proven similarly as in Theorem

Then

If we assume now that E( 2

) jcjW (t) under P ! when min(n 1 ; :::; n d ) ! 1;

(41)

where n = (n 1 ; :::; n d ).

Proof of Example 12.

For this case we take F n = ( u ; u n): Let us note …rst that the variables are square integrable and well de…ned. We also have

For this case we consider the sigma algebras as in Example 12. We start from the following estimate

Since by our conditions c u;u = 0; we obtain

The …rst result of this theorem follows by applying the …rst part of Theorem 7 via Remark 11.

On the other hand, by a moment inequality for U -statistics based on the decoupling procedures, (see Relation 3.1.3. in [START_REF] Giné | Exponential and moment inequalities for U-statistics[END_REF], we obtain for

where ( 0 n ) n2Z d in an independent copy of ( n ) n2Z d . We are now in the position to apply Rosenthal inequality, given in relation 21.5 in [START_REF] Burkholder | Distribution function inequalities for martingales[END_REF], and obtain

Note that, by (42), we have sup u;v 0;j 1 jd u;v (j)j < 1: Also, because u and 0 v are independent and identically distributed, by the properties of '; we can …nd positive constants such that

It remains to note that condition (42) implies condition (38) and then to apply the second part of Theorem 7 and Remark 11.

Remark 14

In Examples 12 and 13 the innovations are i.i.d. …elds. However, the property (2) for the …ltration is a Markovian property and it is not restricted to …ltrations generated by independent random variables. For example, we can