Jean-Pierre Conze
email: conze@univ-rennes1.fr

ON THE QUENCHED FUNCTIONAL CLT IN RANDOM SCENERIES

Keywords: quenched functional central limit theorem, Z d -action, random walk in random scenery, self-intersections of a r.w., toral automorphisms, exponential mixing, ows on homogeneous spaces, S-unit, cumulant

We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random eld (r.f.) along a Z d -random walk in dierent frameworks: probabilistic (when the r.f. is i.i.d. or a moving average of i.i.d. random variables) and algebraic (when the r.f. is generated by commuting automorphisms of a torus or by commuting hyperbolic ows on homogeneous spaces). 37A30.

Introduction

Let G be a group acting on a probability space (E, B, µ) by a measure preserving action (g, x) ∈ G × E → T g x ∈ E. A random walk (Z n ) dened on a probability space (Ω, P) with values in the group G induces a random walk on E. For f ∈ L 2 (E, B, µ), we can consider the sums along the random walk:

n-1 k=0 f (T Z k x).

This general framework leads in practice to dierent situations and methods in the proof of a central limit theorem (CLT) and a functional central limit theorem (FCLT) along the paths of the random walk. In particular the proof of the tightness for the FCLT requires specic tools which it seems interesting to present in examples.

A rst situation is that of a random walk in random sceneries (cf. [START_REF] Kesten | A Limit Theorem Related to a New Class of Self Similar Processes[END_REF], [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF]). For d ≥ 1, let X = (X ) ∈Z d be a strictly Z d -stationary real random eld (r.f.). One can assume that the r.v.s X are dened on a probability space (E, B, µ) on which commuting measure preserving maps T 1 , ..., T d act in such a way 1 that X = T X 0 .

Conversely, given commuting measure preserving invertible maps T 1 , ..., T d and a measurable f on a probability space (E, B, µ), (T f ) ∈Z d is a strictly Z d -stationary random eld. If (Z n )

is a random walk in the group Z d , then the sums along Z n read n-1 k=0 X Z k = n-1 k=0 T Z k X 0 , or n-1 k=0 T Z k f . When (X ) is a d-dimensional random eld of i.i.d. random variables, we obtain the classical random walk in random sceneries.

Another kind of examples in the algebraic case can be obtained as follows. Suppose that G = SL(ρ, Z) and that (E, B, µ) is the torus T ρ , ρ ≥ 2, endowed with the Borel σ-algebra and the Lebesgue measure. The map x → Ax, where A is a matrix in SL(ρ, Z), denes an automorphism of T ρ which preserves µ. When a spectral gap property is available for the transition operator associated to the random walk on G, the previous sums for f in a convenient class of observables satisfy a CLT (cf. [START_REF] Ayyer | Quenched CLT for random toral automorphism[END_REF], [START_REF] Conze | Quenched central limit theorem for random walks with a spectral gap[END_REF]) for P-a.e. ω ∈ Ω.

In the commutative case there is no spectral gap and we use a method based on cumulants.

A rst example considered here is the action of commuting matrices in SL(ρ, Z) acting on T ρ , for which we prove for P-a.e. ω a functional CLT, extending previous results in [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]. A second algebraic example comes from commuting ows on homogeneous spaces. Based on the exponential mixing of all orders proved in [START_REF] Björklund | Quantitative multiple mixing[END_REF], a CLT has been shown in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF] for ergodic sums on Følner sets when the observables are smooth. Likewise we prove here a CLT and its functional version for the sums along a random walk.

The result, a functional CLT for the models described above, is presented for a general aperiodic random walk in dimension d > 1 with a moment of order 2, but the detailed proofs are given in the case of a centered 2-dimensional r.w. The proofs can be adapted easily to the case of transient random walks. We say also some words in the i.i.d. case, when the usual random walk is replaced by a plane Lorentz process generated by a periodic billiard with dispersive obstacles (cf. [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF], [START_REF] Pène | Self-intersections of trajectories of the Lorentz process[END_REF]).

Beyond the CLT, tightness is a main step in the proof of a FCLT. To show it we use the method based on the maximal inequality for associated r.v.s due to Newman and Wright [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF] 1 Underlined letters represent elements of Z d or T d . We write for ( 1 , ..., d ) and T for T or, in the algebraic case, the method based on norm estimates for the maximum of partial sums (cf. Billingsley [START_REF] Billingsley | Convergence of probability measures, 2d edition[END_REF], Móricz [START_REF] Móricz | Moment inequalities and the strong laws of large numbers[END_REF]). A diculty which occurs is that the estimates available for the random walk involve constants depending on the trajectory.

The content of the paper is the following. Section 1 contains results on the variance of sums along a random walk. The independent case is presented in Section 2. Some facts on cumulants are recalled in Section 3, then applied to moving averages in Section 4 and to the algebraic models in Sections 6 and 7. For the tightness in the latter cases, we use the method of maximum of partial sums in an adapted version presented in Section 5.

This paper extends a previous preprint [START_REF] Cohen | On the quenched functional CLT in 2d-random sceneries, examples[END_REF]. We have added the FCLT along a random walk for ows on homogeneous spaces, using the recent results in [START_REF] Björklund | Quantitative multiple mixing[END_REF] and [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF] on the multiple mixing and on the CLT for group actions which are exponentially mixing of all orders. We have also added some remarks about the non nullity of the variance, in particular the observation that there is no degeneracy for the sums along a transient random walk.

Acknowledgements. This research started during visits of the rst author to the IRMAR at the University of Rennes 1 and of the second author to the Center for Advanced Studies in Mathematics at Ben Gurion University. The authors are grateful to their hosts for their support. They thank B. Bekka, Y. Guivarc'h and M. Lin for helpful discussions.

1. Summation along a r.w. and variance 1.1. Random walks and sums along random walks.

First we recall some denitions and results about the random walks on Z d (see [START_REF] Spitzer | Principles of random walk[END_REF], details on the results recalled here can also be found in [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]).

Let (ζ i ) i≥0 be a sequence of i.i.d. random vectors on a probability space (Ω, P) with values in Z d and common probability distribution ν. The associated random walk (r.w.) Z = (Z n ) in Z d starting from 0 is dened by Z 0 := 0, Z n := ζ 0 + ... + ζ n-1 , n ≥ 1.

The r.v.s ζ i can be viewed as the coordinate maps on (Ω, P) obtained as (Z d ) Z equipped with the product measure ν ⊗Z and with the shift θ acting on the coordinates. We have ζ i = ζ 0 • θ i and the cocycle relation

Z n+n = Z n + Z n • θ n , ∀n, n ≥ 0.
Let denote by S := { ∈ Z d : P(ζ 0 = ) > 0} the support of ν and by L the sub-lattice of Z d generated by S. Without loss of generality, we can assume that Z is reduced which means that L is conite in Z d . Therefore the vector space generated by L is R d and d is the `genuine' dimension of the random walk Z.

For simplicity, we will assume that L = Z d (then the random walk Z is said to be aperiodic).

Observe that one can replace a reduced r.w. Z by an aperiodic one, again without loss of generality.

Let D be the sub-lattice of Z d generated by { -, , ∈ S}. We denote by Γ 1 the annulator in T d of D, that is the closed subgroup of T d dened by {t ∈ T d : e 2πi r,t = 1, ∀r ∈ D} and by dγ 1 the Haar probability measure of the group Γ 1 . The r.w. is said to be strictly (or strongly) aperiodic, if D = Z d . For example, the r.w. on Z dened by P(ζ 0 = ±1) = 1 2 is aperiodic, but not strictly aperiodic.

Sums along random walks

Given a strictly Z d -stationary random eld X = (X ) ∈Z d , where the real random variables X are dened on a probability space (E, B, µ), the process of `ergodic sums' along the random walk

(Z n ) is S ω,X n (x) = S ω n (x) := n-1 k=0 X Z k (ω) (x), n ≥ 1, ω ∈ Ω. (1)
If the random eld is represented as X = T f , where T 1 , ..., T d are commuting measure preserving maps and f ∈ L 2 (E, B, µ), the sums read:

S ω n f = n-1 k=0 T Z k (ω) f = ∈Z d w n (ω, ) T f, (2)
where w n (ω, ) (denoted also by w ω n ( )) is the local time of the random walk at time n ≥ 1:

w n (ω, ) = #{k < n : Z k (ω) = } = n-1 k=0 1 Z k (ω)= .
(

) 3
Summing along the random walk amounts to take the ergodic sums for the skew product

(ω, x) → T ζ 0 (ω, x) = (θω, T ζ 0 (ω) x) on Ω × E. Putting F f (ω, x) = F (ω, x) = f (x)
for an observable f on E, we get that the ergodic sums of F for T ζ 0 read:

S n F (ω, x) = n-1 k=0 F (T k ζ 0 (ω, x)) = n-1 k=0 f (T Z k (ω) x) = (S ω n f )(x). (4)
A limit theorem in distribution for the sums S ω n f (with respect to the measure µ on E) obtained for P-a.e. ω is sometimes called quenched. We will use this terminology 2 . If the random variables S n F (ω, x) are viewed as dened on Ω × E endowed with the probability P × µ, a limit theorem under P × µ for theses sums is called annealed.

1.2. Variance for quenched processes.

Let f be a function in L 2 (E, B, µ) with real values. Everywhere we assume (or prove) the absolute summability of the series of decorrelations

∈Z d | X T f f dµ| < ∞, (5)
which implies existence and continuity of the spectral density, the even function given by

ϕ f (t) = ∈Z d T f f e 2πi ,t . (6)
The computation of the variance

E | n-1 k=0 T Z k (ω) f | 2 dµ
, is related to the number of selfintersections of the random walk at time n ≥ 1: 2 We follow here the terminology of [START_REF] Ayyer | Quenched CLT for random toral automorphism[END_REF] used in several papers. The term `quenched' is also used in the random scenery when a limit theorem is shown for the distribution with respect to ω, conditionally to the scenery X.

V n (ω) := #{0 ≤ u, v < n : Z u (ω) = Z v (ω)} = ∈Z d w n (ω, ) 2 = T d | ∈Z d w n ( ) e 2πi ,t | 2 dt. (7)
Let us consider the kernels (which are even functions)

K(w ω n )(t) = | n-1 k=0 e 2πi Z k (ω),t | 2 = | ∈Z d w n (ω, )e 2πi ,t | 2 , K(w ω n )(t) = V n (ω) -1 K(w ω n )(t). (8)
We say that the summation along the r.w. Z is ξ-regular, where ξ is a probability measure on T d , if (for P-a.e. ω) the normalised kernel ( K(w ω n )) n≥1 converges weakly to ξ when n tends to innity, i.e., lim n→∞ T d K(w ω n ) ϕ dt = ξ(ϕ) for every continuous function ϕ on T d . This property is equivalent to (for P-a.e. ω):

lim n→∞ T d K(w ω n )(t) e -2πi p,t dt = lim n→∞ T d K(w ω n )(t) cos(2π p, t ) dt = ξ(p), ∀p ∈ Z d . (9)
Another equivalent formulation is

lim n→∞ V n (ω, p) V n (ω) = ξ(p), ∀p ∈ Z d , for a.e. ω, with V n (ω, p) := #{0 ≤ u, v < n : Z u (ω) -Z v (ω) = p}, p ∈ Z d . ( 10
)
Using the spectral theorem, if f satises (5), ξ-regularity implies that the (asymptotic) normalised variance is, for a.e. ω,

σ 2 (f ) := lim n ∈Z d w n ( , ω) T f 2 2 ∈Z d |w n ( , ω)| 2 = lim n T d K(w ω n )(t) ϕ f (t) dt = ξ(ϕ f ). (11)
It can be shown that the summation along any random walk in Z d is ξ-regular for some measure ξ (cf. [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]).

After some reminders on recurrence/transience, we summarize below the results on the asymptotic variance.

1.2.1. Recurrence/transience.

(cf. [START_REF] Chung | On the distribution of values of sums of random variables[END_REF], [START_REF] Chung | On the recurrence of sums of random variables[END_REF], [START_REF] Spitzer | Principles of random walk[END_REF])

Recall that a r.w.

Z = (Z n ) is recurrent if ∞ n=1 P(Z n = 0) = +∞
P(Z n = 0 innitely often) = 0.
The random walk is said to have a moment of order 2 if

∈Z d P(ζ 0 = ) 2 < ∞.
Assume that Z is reduced in dimension d with a moment of order 2 (for d = 1, a moment of order 1 suces). Then for d=1, 2, Z is recurrent if and only if it is centered. For d ≥ 3, it is always transient.

We denote by Ψ(t) = E[e 2πi ζ 0 ,t ], t ∈ T d , the characteristic function of the r.w.

Observe that Ψ(t) = 1 for t = 0 in T d , since Z is aperiodic. We put

Φ(t) = 1 -|Ψ(t)| 2 |1 -Ψ(t)| 2 for t = 0, = 0 for t = 0. (12)
Remark 1.1. Φ is nonnegative and Φ(t) = 0 only on Γ 1 . Hence, using again aperiodicity, it is positive for a.e. t, except when the r.w. is `deterministic' (i.e., if P(ζ 0 = ) = 1 for some ∈ Z d , so that |Ψ(t)| ≡ 1 in this case).

A r.w. of genuine dimension d which is aperiodic is transient or recurrent depending on whether e( 1 1-Ψ ) is integrable or not on the d-dimensional unit cube ( [START_REF] Spitzer | Principles of random walk[END_REF]).

Transient case

In the transient case, one can show:

Theorem 1.2. ([27]) Let Z = (Z n ) be a transient aperiodic random walk in Z d .
a) The function Φ is integrable on T d and, with a nonnegative constant K, we have

I( ) := 1 =0 + ∞ k=1 [P(Z k = ) + P(Z k = -)] = T d cos(2π , t ) Φ(t) dt + K, ∀ ∈ Z d . b 1 ) Suppose d = 1. If m 1 (Z) = +∞, then K = 0. If m 1 (Z) < ∞, then Z is non centered (because it is transient) and K = | ∈Z P(ζ 0 = ) | -1 . b 2 ) If d > 1, then K = 0.
c) Denoting by dξ 0 (t) the measure Φ(t)dt + Kδ 0 (t), we have, for a.e. ω,

1 n K ω n (t) cos(2π , t ) dt = 1 =0 + 1 n n-1 k=1 n-k-1 j=0 [1 Z k (θ j ω)= + 1 Z k (θ j ω)=-] → n→∞ I( ) = cos(2π , t ) dξ 0 (t). ( 13
)
It follows that the summation along a transient r.w. behaves for the normalisation like the iteration of a single transformation, is ξ 0 -regular (up to a constant factor) and that

lim n 1 n n-1 k=0 T Z k (ω) f 2 2 = Φ(t) ϕ f (t) dt + Kϕ f (0). (14)
From ( 13), ( 14) and the expression of ϕ f , ϕ f (t) = ∈Z d T f, f cos(2π , t ), we deduce:

Φ(t) ϕ f (t) dt + Kϕ f (0) = ϕ f (t) dξ 0 (t) = ∈Z d T f, f cos(2π , t )dξ 0 (t) = ∈Z d I( ) T f, f = f 2 2 + 2 k≥1 ∈Z d P(Z k = ) T f, f . Remark 1.3. (about the variance in the non deterministic transient case) Let f be in L 2 (E, µ)
with real values and satisfying [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF]. Suppose that f is not a.e. equal to 0. a) By Remark 1.1, Φ is positive a.e. Therefore Φ ϕ f dt = 0 implies ϕ f (t) = 0 a.e. which is is not possible since

ϕ f dt = f 2 .
Therefore the (quenched) asymptotic variance, which is given by ( 14), is > 0.

b) Let F f (ω, x) = F (ω, x) = f (x).
For the map T ζ 0 acting on the product space Ω × E endowed with the product measure P × µ, we have

T n ζ 0 F F dµ dP = (E1 Zn= ) T f f = ∈Z d P(Z n = ) T f, f , n ≥ 0.
In the transient case it holds, for every , k≥1 P(Z k = ) ≤ k≥1 P(Z k = 0) < +∞. Therefore the density of the spectral measure for F f and the map T ζ 0 is

f 2 2 + 2 k≥1 ∈Z d P(Z k = ) T f, f cos 2πkt.
The asymptotic variance lim n

1 n n-1 k=0 T k ζ 0 F f 2 2
for the annealed model is the same as for the quenched model and is equal to

f 2 2 + 2 k≥1 ∈Z d P(Z k = ) T f, f .
It follows that the function F f on Ω × E (which depends only on the second coordinate), with f as above and non a.e. null, is never a coboundary in L 2 (P × µ) for T ζ 0 , because the asymptotic variance is non null. Observe also that F f is not even a measurable coboundary, at least when the CLT holds, which is the case of the situations that we are going to consider here. This follows from the fact that, for a single measure preserving transformation, if an observable is a coboundary in the space of measurable functions, then the limiting distribution of the ergodic sums after normalisation by any sequence tending to innity is the Dirac mass at 0, which is excluded here.

Recurrent case

Let us consider now the case d = 2 and a centered random walk Z with a moment of order 2. By the local limit theorem (LLT), Z is recurrent.

A non standard normalization occurs in the CLT for sums along Z n as recalled below. There are C 0 , C nite positive constants 3 such that (cf. [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF] Lemma 2.6, [START_REF] Lewis | A law of the iterated logarithm for random walk in random scenery with deterministic normalizers[END_REF] Proposition 1.4 for [START_REF] Erdös | Some problems concerning the structure of random walk paths[END_REF] and ( 16), [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF] Theorem 4.13 for (17)):

E(V n ) ∼ C 0 n log n, Var(V n ) ≤ Cn 2 , ( 15
)
ϕ n (ω) := V n (ω) C 0 n log n → 1, for a.e. ω, (16)
ϕ n (ω, p) := V n (ω, p) C 0 n log n → 1, ∀p ∈ Z d , for a.e. ω. (17)
Therefore the summation along the r.w.

Z is δ 0 -regular: the normalised kernel satises lim n K(w ω n )(t) e -2πi p,t dt = 1, ∀p ∈ Z d and the asymptotic variance is

σ 2 (f ) = lim n (C 0 n log n) -1 n k=0 T Z k (ω) f 2 2 = k∈Z d T k f f = ϕ f (0). ( 18
)
3 If the r.w. is strongly aperiodic, C 0 = (π √ det Σ) -1 , where Σ is the covariance matrix of the increments of the r.w.

The results presented below are valid for the cases covered above, excludes only the onedimensional recurrent case. We stress that, in the recurrent 2-dimensional case, the variance can be degenerate, while this does not occur in the transient case unless f = 0.

1.2.2. Number of self-intersections of a 2-dimensional centered r.w.

In this subsection, we study more precisely the case d = 2 and a centered random walk (Z n ) with a moment of order 2.

If I, J are intervals, the quantity V (ω, I, J, p) := u∈I e 2πi Zu(ω),t v∈J e -2πi Zv(ω),t e -2πi p,t dt = #{(u, v)

∈ I × J : Z u (ω) -Z v (ω) = p} (19)
is non negative and increases when I or J increases for the inclusion order.

We write simply V (ω, I, p) if I = J, V (ω, I) for V (ω, I, 0), V n (ω) and V n (ω, p) as above for V (ω, [0, n[) and V (ω, [0, n[, p).

Observe that V (ω, J) = ∈Z 2 w(ω, J, ) 2 , where w(ω, J,

) = i∈J 1 Z i (ω)= . Notice also that V (ω, [b, b + k[) = V (θ b ω, [0, k[) = V k (θ b ω), for b ≥ 0, k ≥ 1. Let A, B be in [0, 1]. We have: 4 V (ω, [nA, nB], p) = ( u∈[nA,nB] e 2πi Zu(ω),t ) ( v∈[nA,nB] e -2πi Zv(ω),t ) e -2πi p,t dt = #{u, v ∈ [0, n(B -A)] : u-1 i=0 ζ 0 (θ i+nA ω) - v-1 i=0 ζ 0 (θ i+nA ω) = p} = V (θ nA ω, [0, n(B -A)], p).
By [START_REF] Esary | Association of Random Variables with Applications[END_REF] and [START_REF] Evertse | Linear equations in variables which lie in a multiplicative group[END_REF] there a set Ω 0 of full probability such that

V n (ω) ≤ K(ω) n log n, ∀n ≥ 2, where the function K ≥ 0 is nite on Ω 0 ,

for any xed A ∈]0, 1], V (ω, [1, nA], p) ∼ C 0 nA log n, for ω ∈ Ω 0 . w n (ω, ) = o(n ε ), for a.e. ω, for every ε > 0.

(

) 22
For a simple r.w. on Z 2 , Erdös and Taylor [START_REF] Erdös | Some problems concerning the structure of random walk paths[END_REF] have shown: lim sup

n sup ∈Z 2 w n (ω, ) (log n) 2 ≤ 1 π .
The result has been extended by Dembo, Peres, Rosen and Zeitouni [START_REF] Dembo | Thick points for planar Brownian motion and the Erdös-Taylor conjecture on random walk[END_REF]. They proved for an aperiodic centered random walk on Z 2 with moments of all orders:

lim n sup ∈Z 2 w n (ω, ) (log n) 2 = 1 π .
Remark 1.4. In Section 7 we will use the upper bound

lim n sup ∈Z 2 w n (ω, ) (log n) 2 ≤ 1 π . ( 23
)
4 For simplicity, in the formulas above and below, we write nA, nB instead of nA or nA + 1, nB , θ t instead of θ t . The equalities are satised up to the addition of quantities which are bounded independently from A, B, n.

As shown in the proof in [START_REF] Dembo | Thick points for planar Brownian motion and the Erdös-Taylor conjecture on random walk[END_REF], it suces for the validity of [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF] to suppose that the 2-dimensional r.w. is aperiodic. Moreover, the proof of the upper bound is based on the local limit theorem which uses only the existence of the moments of order 2.

We will need also to bound, for

1 , 2 , 3 ∈ Z d , W n (ω, 1 , 2 , 3 ) := (24) #{1 ≤ i 0 , i 1 , i 2 , i 3 < n : Z i 1 (ω)-Z i 0 (ω) = 1 , Z i 2 (ω)-Z i 0 (ω) = 2 , Z i 3 (ω)-Z i 0 (ω) = 3 }.
Lemma 1.5. There exists a positive integrable function C 3 such that

W n (ω, 1 , 2 , 3 ) ≤ C 3 (ω) n (log n) 5 , ∀n ≥ 1. (25)
Proof. It suces to bound the sum with strict inequality between indices

W n (ω) = 1≤i 0 <i 1 <i 2 <i 3 ≤n 1 Z i 1 -Z i 0 = 1 .1 Z i 2 -Z i 1 = 2 -1 .1 Z i 3 -Z i 2 = 3 -2 .
Using independence and the local limit theorem for the random walk, we nd the bound

W n (ω) dP(ω) ≤ C 1 i 0 ,i 1 ,i 2 ,i 3 ∈[1,n] (i 1 i 2 i 3 ) -1 ≤ C 2 n (log n) 3 . Therefore ∞ p=1 2 -p (log(2 p )) -5 W 2 p dP < ∞. The function C(ω) := ∞ p=1 2 -p (log(2 p )) -5 W 2 p is
integrable and we have:

W 2 p (ω) ≤ C(ω) 2 p (log(2 p )) 5 , ∀p ≥ 1.
Let p n be such that: 2 pn-1 ≤ n < 2 pn . Since W n is increasing with n, we obtain:

W n (ω) ≤ W 2 pn (ω) ≤ C(ω) 2 pn (log(2 pn )) 5 ≤ C(ω) 2n (log(2n)) 5 ≤ C (ω) n(log n) 5 .

Variance for the nite dimensional distributions

The following lemma will be applied to the successive return times of a point ω into a set under the iteration of the shift θ.

Lemma 1.6. Let (y(j), j ≥ 1) be a sequence with values in {0, 1} such that lim n 1

n n j=1 y(j) = a > 0. If (k r )
is the sequence of successive times such that y(k r ) = 1, then, for every δ > 0, there is n(δ) such that,

for n ≥ n(δ), k r+1 -k r ≤ δn, for all r ∈ [1, n].
Proof. Since r = kr j=1 y(j), we have: k r /r = k r / kr j=1 y(j) → a -1 . Hence, for every δ > 0,

there is n 1 (δ) such that 0 < k r+1 -k r ≤ δr, for r ≥ n 1 (δ). Therefore, if n ≥ n 1 (δ), then 0 < k r+1 -k r ≤ δr ≤ δn, for r ∈ [n 1 (δ), n]. If n(δ) ≥ n 1 (δ) is such that k r+1 -k r ≤ δn(δ) for r ≤ n 1 (δ)
, we get the result of the lemma. Lemma 1.7. Let Λ be a measurable set in Ω of positive measure. Let k r = k r (ω) be the successive times such that θ kr ω ∈ Λ. For ω in a set Ω Λ of full measure, for every positive small enough δ, there is n(δ) such that for n ≥ n(δ)

1) k r+1 -k r ≤ δn, for all r ∈ [1, n]; moreover, k n ∼ cn, with c = P(Λ) -1 , when n → ∞; 2) there are integers v < 2/δ and 0 = ρ (n) 1 < ρ (n) 2 < ... < ρ (n) v ≤ n < ρ (n) v+1 , such that θ ρ (n) i ω ∈ Λ and 1 2 δn ≤ ρ (n) i+1 -ρ (n) i ≤ 3 2 δn, for i = 1, ..., v.
Proof. Since θ is ergodic on (Ω, P), Birkho ergodic theorem implies lim n 1 n n-1 0 1 Λ (θ k ω) = P(Λ) > 0, for a.e. ω and k n /n → P(Λ) -1 . Hence Lemma 1.6 implies 1). For 2), we select in the sequence (k r ) an increasing sequence of visit times to the set Λ satisfying the prescribed conditions by eliminating successive times which are at a distance < 1 2 δn.

Asymptotic orthogonality of the cross terms Proposition 1.8. For 0 < A < B < C < D < 1, p ∈ Z, it holds a.e.

( nB v=nA e 2πi Zv(ω),u ) ( nD w=nC e -2πi Zw(ω),u ) e -2πi p,u du = ε n (ω) n log n, with ε n (ω) → 0. ( 26
)
The above integral is the non negative self-intersection quantity: V (ω, [nA, nB], [nC, nD], p). By ( 19), V (ω, I, J, p) increases when I or J increases. Hence, it suces to show [START_REF] Schlickewei | S-unit equations over number elds[END_REF] for the intervals [1, nA], [nA, n], for 0 < A < 1. The proof below is based on ( 17) and ( 21).

Lemma 1.9. There is a set Ω ⊂ Ω such that P( Ω) = 1 and for all ω ∈ Ω, the following holds:

lim n ϕ nB (θ nA ω, p) = lim n V (ω, [nA, n], p) C 0 nB log n = 1, for A ∈]0, 1[, B = 1 -A; (27)
V (ω, [1, nA], [nA, n], p) + V (ω, [nA, n], [1, nA], p) = ε n (ω) n log n, with ε n (ω) → 0. (28)
Proof. 1) The set Ω. For every L ≥ 1 and δ > 0, let Λ(L, δ)

:= {ω : ϕ n (ω, p) -1 ∈ [-δ, δ], ∀n ≥ L}. We have lim L↑∞ P(Λ(L, δ)) = 1. There is L(δ) such that P(Λ(L(δ), δ)) ≥ 1 2 .
Let (δ j ) be a sequence tending to 0. We apply Lemma 1.7 to Λ(L(δ j ), δ j ) for each j. By taking the intersection of the sets of full measure Ω Λ(L(δ j ),δ j ) , we get a set Ω 1 of full measure. The set Ω is the intersection of Ω 1 with the set Ω 0 (of full measure) for which the law of large numbers holds for (V n (ω)). Let ω ∈ Ω.

2) Proof of [START_REF] Spitzer | Principles of random walk[END_REF]. We have

V (ω, [nA, n[, p) = V (θ nA ω, [0, n(1 -A)[, p) and V (ω, [1, n], p) -V (ω, [1, nA[, p) -V (ω, [nA, n], p) = V (ω, [1, nA[, [nA, n[, p) + V (ω, [nA, n], [1, nA[, p) ≥ 0. (29)
Claim: for an absolute constant C 1 depending on A and p, for every δ, for n big enough,

ϕ nB (θ nA ω, p) = V (ω, [nA, n], p) C 0 n(1 -A) log n ∈ [1 -C 1 δ, 1 + C 1 δ]. (30)
Let us show the claim. We put B = 1 -A and take δ ∈]0, B[.

Upper bound:

The law of large numbers for V n (ω, p) implies, with

|ε n |, |ε n | ≤ δ for n big enough, C -1 0 V (ω, [1, n], p) = (1 + ε n ) n log n, C -1 0 V (ω, [1, nA], p) = (1 + ε n ) nA log n.
This implies by ( 29)

V (ω, [nA, n], p) C 0 nB log n ≤ (1 + ε n ) n log n -(1 + ε n ) nA log n nB log n ≤ 1 + |ε n | B + |ε n |A B ≤ 1 + 1 + A B δ.
Lower bound: We apply Lemma 1.7 to Λ(L(δ), δ). Let n A , n A be two consecutive visit times ≤ n such that n A ≤ nA < n A . For n big enough, we have 0 < n A -n A ≤ δn and

n A = nA (1 -ρ n ), n A = nA (1 + ρ n ), with 0 ≤ Aρ n , Aρ n ≤ δ.
Since ω ∈ Ω, we have for n big enough, with |δ n | ≤ δ,

C -1 0 V (ω, [n A , n], p) ≥ (1 -δ n )(n -n A ) log(n -n A ) = (1 -δ n )(nB -nAρ n ) log(nB -nAρ n ).
It follows, for δ (hence ρ n ) small:

V (ω, [n A , n], p) C 0 (1 -δ n ) nB log(nB) ≥ (nB -nAρ n ) log(nB -nAρ n ) nB log(nB) = (B -Aρ n ) [log(nB) + log(1 -A B ρ n )] B log(nB) ≥ (1 - A B ρ n ) -2(1 - A B ρ n ) A B ρ n log(nB) ≥ 1 - A B ρ n -2 A B ρ n log(nB) ≥ 1 -B -1 δ(1 + 2 log(nB)
).

As V (ω, J, p) increases when the set J increases, we have by the choice of n A and n A :

V (ω, [n A , n], p) ≤ V (ω, [nA, n], p).
Therefore, for n such that log(nB) ≥ 2, we have

V (ω, [nA, n], p) C 0 nB log(nB) ≥ (1 -δ) (1 - 2 B δ) ≥ 1 -δ(1 + 2 B
).

This shows the lower bound. Altogether with the upper bound, this proves the claim (30).

3) Proof of (28). Let δ > 0. According to (29) and (30), for n big enough, we have with

|ε n | ≤ C 1 δ: V (ω, [1, nA], [nA, n], p) + V (ω, [nA, n], [1, nA], p) = V (ω, [1, n], p) -V (ω, [1, nA], p) -V (ω, [nA, n], p) = C 0 [(1 + ε n ) n log n -(1 + ε n ) nA log n -(1 + ε n ) n(1 -A) log n ≤ (2 + C 1 ) C 0 δ n log n.
Let a 1 , ..., a s be real numbers and

0 = t 0 < t 1 < ... < t s-1 < t s = 1 a subdivision of [0, 1].
For the asymptotic variance of s j=0 a j nt j-1 ≤k<nt j T Z k (ω) f , which is used later, we need the following lemma.

Notation: With the convention of footnote 4, we will write nt j k=nt j-1 instead of nt j-1 ≤k<nt j .

Recall that f has a continuous spectral density ϕ f . Lemma 1.10. For a.e. ω and for every partition (t j ), we have

(C 0 n log n) -1 s j=1 a j nt j k=nt j-1 T Z k (ω) f 2 2 → ϕ f (0) s j=1 a 2 j (t j -t j-1 ). (31)
Proof. 1) Recall that proving (31) amounts to prove

(C 0 n log n) -1 | s j=1 a j nt j k=nt j-1 e 2πi Z k (ω),u | 2 ϕ f (u) du → ϕ f (0) s j=1
a 2 j (t j -t j-1 ).

1) First suppose that ϕ f is a trigonometric polynomial ρ, which allows to use [START_REF] Schlickewei | S-unit equations over number elds[END_REF] for a nite set of characters e -2πi p,u . Using [START_REF] Kesten | A Limit Theorem Related to a New Class of Self Similar Processes[END_REF] for the asymptotic variance starting from 0, we have

(C 0 n log n) -1 tn k=0 T Z k (ω) f 2 2 → tρ(0), for t ∈]0, 1[. By Lemma 1.9, (C 0 n log n) -1 tn k= sn T Z k (ω) f 2 2 → (t -s) ρ(0), for 0 < s < t < 1.
Expanding the square and using that the cross terms are asymptotically negligible, we have

(C 0 n log n) -1 | s j=1 a j nt j k=nt j-1 e 2πi Z k (ω),u | 2 ρ(u) du ∼ (C 0 n log n) -1 s j=1 a 2 j | nt j k=nt j-1 e 2πi Z k (ω),u | 2 ρ(u) du → ρ(0) s j=1 a 2 j (t j -t j-1 )
.

This shows (31) for trigonometric polynomials.

2) For a general continuous spectral density ϕ f , for ε > 0, let ρ be a trigonometric polynomial,

such that ϕ f -ρ ∞ < ε. Remark that | s j=1 a j nt j k=nt j-1 e 2πi Z k (ω),u | 2 du ≤ s j,j =1 a j a j V (ω, [nt j-1 , nt j ], [nt j -1 , nt j ], 0) ≤ ( s j=1 |a j |) 2 V n (ω).
Therefore we have:

(C 0 n log n) -1 | s j=1 a j nt j k=nt j-1 e 2πi Z k (ω),u | 2 ϕ f (u) du -ϕ f (0) s j=1 a 2 j (t j -t j-1 ) ≤ (C 0 n log n) -1 | s j=1 a j nt j k=nt j-1 e 2πi Z k (ω),u | 2 ρ(u) du -ρ(0) s j=1 a 2 j (t j -t j-1 ) +ε [(C 0 n log n) -1 | s j=1 a j nt j k=nt j-1 e 2πi Z k (ω),u | 2 du + s j=1 a 2 j (t j -t j-1 )].
By the remark, the above quantity inside [ ] is less than

( s j=1 |a j |) 2 (C 0 n log n) -1 V n (ω) + s j=1 a 2 j (t j -t j-1 )
, which is bounded uniformly with respect to n by [START_REF] Leonov | On the central limit theorem for ergodic endomorphisms of compact commutative groups[END_REF]. Therefore we can conclude for a general continuous spectral density by step 1).

Remarks 1.11. 1) In Lemma 1.7, the dynamical system (Ω, θ, P) can be replaced by any ergodic dynamical system.

2) If the spectral density is constant (i.e., when the X k 's are pairwise orthogonal), ( 26) and

(31) are a consequence of the law of large numbers for the number of self-intersections, that is Vn(ω) C 0 n log n → 1. The law of large numbers for V n (ω, p), p = 0, is not needed.

3) A result analogous to Proposition 1.8 is valid if the r.w. Z is transient: for

0 < A < B < C < D < 1, p ∈ Z, ( nB v=nA e 2πi Zv(ω),u ) ( nD w=nC e -2πi Zw(ω),u ) e -2πi p,u du = ε n (ω) n, with ε n (ω) → 0.
(32) 1) Convergence of the nite dimensional distributions:

∀ 0 = t 0 < t 1 < ... < t r = 1, (Y n (t 1 ), ..., Y n (t r )) =⇒ n→∞ (W t 1 , ..., W tr ),
a property which follows (by the Cramér-Wold device) from

r j=1 a j (Y n (t j ) -Y n (t j-1 )) =⇒ N (0, r j=1 a 2 j (t j -t j-1 )), ∀(a j ) 1≤j≤r ∈ R. (33)
2) Tightness of the process: The condition of tightness reads:

∀ε > 0, lim δ→0 + lim sup n µ(x ∈ E : sup |t -t|≤δ |Y n (x, t ) -Y n (x, t)| ≥ ε) = 0. (34)
Let (Z n ) be a random walk on (Ω, P) with values in Z d , d ≥ 1, and let X = (X (x)) ∈Z d = (T f (x)) ∈Z d be a d-dimensional random eld dened on a probability space (E, B, µ). We take sums along Z n , introducing so another alea ω (which will be xed in the quenched setting).

A quenched FCLT is satised by the sums along Z n if, for P-a.e. ω, the functional central limit theorem holds for the process S ω,X [nt] (x) (cf. Notation (1)) after normalisation. For d = 2, it means that, for P-a.e. ω, the functional central limit theorem holds for the normalised process

(Y n (ω, x, t)) t∈[0,1] := S ω,X [nt] (x) √ C 0 n log n t∈[0,1] . ( 35
)

Random walk in random scenery

We consider in this section d = 2 and the random walk in random scenery S ω,X n (x), that is the process (Y n ) dened by (35) when X is a 2-dimensional random eld of i.i.d. real variables with E(X 2 0 ) = 1 and mean 0 on a probability space (E, B, µ). As the random eld is i.i.d centered with E(X 2 0 ) = 1, the corresponding spectral density is constant and identically = 1.

It was shown by E. Bolthausen [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF] that this process satises an annealed FCLT: with respect to the probability P × µ, the law of Y n converges weakly to the Wiener measure.

We show a quenched FCLT for the r.f. X (and when X is a r.f. of moving averages of i.i.d. random variables in Section 4). As for the annealed FCLT in [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF] and for Theorem 2.2 in [START_REF] Deligiannidis | Computation of the asymptotics of the variance of the number of selfintersections of stable random walks using the Wiener-Darboux theory[END_REF] for a 1-dimensional stable r.w., the proof of Theorem 2.2 below is based on the maximal inequality shown by Newman and Wright [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF] for associated r.v.s. Denition 2.1. (cf. [START_REF] Esary | Association of Random Variables with Applications[END_REF]) Recall that real random variables X 1 , . . . , X n are associated if, for every n ≥ 1, for all non-decreasing (in each coordinate) functions f, g :

R n → R, we have Cov(f (X 1 , . . . , X n ), g(X 1 , . . . , X n )) ≥ 0 (if the covariance exists).
Non-decreasing functions of a family of associated random variables are associated [START_REF] Esary | Association of Random Variables with Applications[END_REF]. Independent variables are associated. A family made out of a family of associated (in particular independent) variables with possible repetition is associated.

It follows that, if (X , ∈ Z 2 ), are associated r.v.s, in particular independent, then the r.v.s (X Z k (ω) , k ≥ 0), are associated for every ω ∈ Ω.

Theorem 2.2. If E(X 2 0 ) = 1, for P-a.e. ω, the process Y n (ω, x, t t∈[0,1] = S ω,X nt (x) √ n log n t∈[0,1]
satises a FCLT with asymptotic variance σ 2 = (π

√ det Σ) -1 .
Proof. 1) For the convergence of the nite dimensional distributions, the proof, relying on Cramér-Wold's theorem and Lindeberg's CLT, is as in [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF]. Another proof, based on truncation and cumulants, can be given, like for the more general case of moving averages in Section 4.

2) Tightness of the process (Y n )

The following is shown in [23, p. 673]:

Let U 1 , U 2 , . . . be centered associated random variables with nite second order moment. Put S k = k j=1 U j , for k ≥ 1. Then, for every λ > 0 and n ≥ 1, we have

µ( max 1≤k≤n |S k | ≥ λ S n 2 ) ≤ 2µ |S n | ≥ (λ - √ 2) S n 2 . ( 36
)
Inequality (36) can be applied for every xed ω to U j = X Z j (ω) and to the sums

S J = b+k j=b X Z j (ω) for any interval J = [b, b + k] ⊂ [0, n]. We also note that E(S 2 J ) = X 0 2 2 V (ω, J). a) First, let us assume that E(X 4 0 ) < ∞. With K(ω)
given by (20), we have i∈J

X Z i (ω) 4 4,µ = 3E(X 2 0 ) 2 1 = 2 w(ω, J, 1 ) 2 w(ω, J, 2 ) 2 + E(X 4 0 ) w(ω, J, ) 4 ≤ 4 E(X 4 0 ) V (ω, J) 2 ≤ 4 E(X 4 0 ) (K(θ b ω)) 2 (k log k) 2 . ( 37
)
Let C 1 be a constant > 0 such that P{ω : K(ω) ≤ C 1 } > 0. Using Lemma 1.7, for δ ∈]0, 1[ and n big enough, there are times 0 = 20) and (37), with C = (2C 1 ) 1 2 , we have:

ρ 1 < ρ 2 < ... < ρ v ≤ n < ρ v+1 , with v < 2/δ, such that K(θ ρ i ω) ≤ C 1 and 1 2 δn ≤ ρ i+1 -ρ i ≤ 3 2 δn, for i = 1, . . . , v. Let t i = ρ i n , λ = ε √ δ , J i = [ρ i-1 , . . . , ρ i [, m i = 2 3 (ρ i+1 -ρ i ) ≤ δn. By (
ρ i j=ρ i-1 X Z j (ω) 2 ≤ C X 0 2 (n δ log(nδ)) 1 2 , ( 38
)
ρ i j=ρ i-1 X Z j (ω) 4 ≤ C X 0 4 (n δ log(nδ)) 1 2 , ∀i. (39)
We can assume that δ ≤ 1 8

ε 2 C 2 X 0 2 2 . This implies the inequality λ i - √ 2 ≥ 1 2 λ i used below.
Using (36), we get (with σ (i) :=

ρ i j=ρ i-1 X Z j (ω) 2 and λ i := ε √ n log n/σ (i) )
, by Chebyshev's inequality (for a moment of order 4) and (39):

µ( sup

t i-1 ≤s≤t i | sn j=[t i-1 n X Z j (ω) | ≥ ε n log n) = µ( max ρ i-1 ≤k≤ρ i | k j=ρ i-1 X Z j (ω) | ≥ λ i σ (i) ) ≤ 2µ(| ρ i j=ρ i-1 X Z j (ω) | ≥ (λ i - √ 2) σ (i) ) ≤ 2µ(| ρ i j=ρ i-1 X Z j (ω) | ≥ 1 2 λ i σ (i) ) ≤ 2µ(| ρ i j=ρ i-1 X Z j (ω) | ≥ 1 2 ε n log n) ≤ 2 C 4 X 0 4 4 (n δ log(nδ)) 2 1 16 ε 4 (n log n) 2 ≤ 32 C 4 X 0 4 4 δ 2 ε 4 . (40)
Observe now that (cf. [2, Theorem 7.4, p. 83])

µ( sup

|t -t|≤δ |Y n (t) -Y n (s)| ≥ 3ε) ≤ v i=1
µ( sup

t i-1 ≤s≤t i | sn j=[t i-1 n X Z j (ω) | ≥ ε C 0 n log n).
Hence, by (40) we get µ( sup

|t -t|≤δ |Y n (t) -Y n (s)| ≥ 3ε) ≤ 32 C 4 C 2 0 X 0 4 4 2 δ δ 2 ε 4 = 64 C 4 C 2 0 X 0 4 4 δ ε 4 .
b) Now we use a truncation. For L > 0, let

XL k := X k 1 {|X k |≤L} -E(X k 1 {|X k |≤L} ), XL k := X k -XL k = X k 1 {|X k |>L} -E(X k 1 {|X k |>L} ), Ŷ L n (t) = 1 √ C 0 n log n tn j=0 XL Z j (ω) and Ỹ L n (t) := Y n (t) -Ŷ L n (t) = 1 √ C 0 n log n tn j=0 XL Z j (ω) .
Since we have still sums of associated random variables, all what we have done above (including (36)) holds for both sums, except that for the unbounded part of the truncation we only have a moment of order 2. We use Chebyshev's inequality (for a moment of order 2) and (38) to control the unbounded truncated part:

µ(| ρ i j=ρ i-1 XL Z j (ω) | ≥ 1 2 ε n log n) ≤ C 2 XL 0 2 2 n δ log(nδ) 1 4 ε 2 n log n ≤ 4C 2 XL 0 2 2 δ ε 2 .
Hence, for n large, the sum over i of the above measures at left is comparable for some constant

C to C v i=1 XL 0 2 2 ≤ C δ δ XL 0 2 2 ε 2 = C XL 0 2 2 ε 2 .
Applying the inequality µ(|f

+ g| ≥ ε) ≤ µ(|f | ≥ ε 2 ) + µ(|g| ≥ ε 2 ) to Y n (t) = Ŷ L n (t) + Ỹ L n (t), we
obtain the bound:

µ( sup |t -t|≤δ |Y n (t ) -Y n (t)| ≥ 3ε) ≤ 2 10 C 4 C 2 0 L 4 δ ε 4 + 4C XL 0 2 2 ε 2 .
We need, for xed ε > 0, lim δ→0

+ lim sup n µ(sup |t -t|≤δ |Y n (t ) -Y n (t)| ≥ 3ε) = 0. Let η > 0. First we take L such that 4C XL 0 2 2 ε 2 < 1 2 η, then δ such that 2 10 C 4 C 2 0 L 4 δ ε 4 ≤ 1 2 η.

A model based on the Lorentz process

We sketch briey how to obtain a version of a FCLT when the random walk is replaced by the movement of a particle in a dispersing periodic billiard. We refer to [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF] and [START_REF] Pène | Self-intersections of trajectories of the Lorentz process[END_REF] for more details on this model.

Let be given a billiard table in the plane, union of Z 2 -periodically distributed obstacles with pairwise disjoint closures. We consider a point particle moving in the complementary Q of the billiard table in R 2 with unit speed and elastic reection o the obstacles. By sampling the ow at the successive times of impact with the obstacles, we obtain a Poincaré's section of the billiard ow, the billiard transformation.

We assume that the obstacles are strictly convex with pairwise disjoint closures and boundaries of class C r+1 with curvature > 0 (Sinai's billiard or Lorentz's process). Moreover we make the hypothesis of nite horizon (the time between two subsequent reections is uniformly bounded).

Suppose that to each obstacle is associated a real random variable with zero expectation, positive and nite variance, independent of the motion of the particle and that the family of these r.v.s is i.i.d.

Like in an innite pinball with random gain, at each collision with an obstacle, the particle wins the amount given by the random variable associated with the obstacle which is met. Let

W n be the total amount won by the particle after n reections. An annealed FCLT for W n has been shown by F. Pène ( [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF]): there exists β 0 > 0 such that W [nt] β 0 n log n converges weakly to the standard Wiener process.

In order to prove a quenched version, we use [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF]Proposition 7], in place of [START_REF] Móricz | Moment inequalities and the strong laws of large numbers[END_REF] for the r.w., and [25, Corollary 4] (the main and most dicult step), which gives for the self-intersections of the billiard transformation a law of large numbers replacing [START_REF] Esary | Association of Random Variables with Applications[END_REF]. Then, by 1) in Remarks 1.11 and by the preceding method for the r.w. in random sceneries, we obtain the quenched version of the FCLT for this model.

Cumulants and CLT

For the models of random elds in Sections 4, 6, 7, we need to recall some tools, in particular the method of cumulants which can be used to prove a CLT for dynamical systems satisfying a mixing property of all orders.

In 1960, Leonov ([19], [START_REF] Leonov | On the central limit theorem for ergodic endomorphisms of compact commutative groups[END_REF]) applied it to a single algebraic endomorphism of a compact abelian group. In [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF], [START_REF] Cohen | Almost mixing of all orders and CLT for some Z d -actions on subgroups of F Z d p[END_REF], it was applied to multidimensional actions given by algebraic endomorphisms in the connected case and in some non connected cases. Recently the method of cumulants has been used in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF] to prove a CLT for multiple mixing actions with exponential rate. Using the tightness criterium given in Section 5, we will obtain a functional version in these examples.

Moments and cumulants.

For r ≥ 1, let X 1 , ..., X r be r real centered bounded random variables. We denote by J r the set {1, ..., r}. For any subset I = {i 1 , ..., i p } ⊂ J r , we put m(I)

:= E(X i 1 ...X ip ). The cumulant of order r is C(X 1 , ..., X r ) = Q={I 1 ,...,Ip}∈Q (-1) p-1 (p -1)! m(I 1 ) • • • m(I p ). (41) Putting s(I) := C(X i 1 , ..., X ip ) for I = {i 1 , ..., i p }, we have E(X 1 • • • X r ) = m(J r ) = Q={I 1 ,...,Ip}∈Q s(I 1 ) • • • s(I p ), (42)
where in both formulas, Q is the set of partitions Q = {I 1 , I 2 , ..., I p } of J r into p ≤ r nonempty subsets, with p varying from 1 to r.

E(Y 4 ) = 3E(Y 2 ) 2 + C (4) (Y ). (43)
In the next sections, we are going to consider random elds obtained by a measure preserving action (T h , h ∈ H) of a group H (with a left-invariant metric d on H) on a probability space (E, B, µ). If f be a measurable bounded centered function on (E, µ), the composed function f • T h is denoted by T h f . For H = (h 1 , ..., h r ) in H r , we can apply the denition of moments and cumulants to (T h 1 f, ..., T hr f ). Notation 3.1. For the purpose of Section 7, we introduce some notations as in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF]. Let H be in H r and I, J non empty subsets of J r . We set

d r (H) := max i,j d(h i , h j ), d I (H) := max i,j∈I d(h i , h j ), d I,J (H) := min i∈I, j∈J d(h i , h j ) Let Q be a partition of J r , with |Q| ≥ 2. We set d Q (H) := max I∈Q d I (H), d Q (H) := min I =J, I,J∈Q d I,J (H),
and, for 0 ≤ α < β,

∆(β) := {H ∈ H r : d r (H) ≤ β}, ∆ Q (α, β) := {H ∈ H r : d Q (H) ≤ α and d Q (H) > β}.
The elements of a conguration H = (h 1 , h 2 , ..., h r ) in ∆(β) can be viewed as `clustered', since (h 1 , h 2 , ..., h r ) ∈ ∆(β) implies (h 2 , ..., h r ) ⊂ B d (h 1 , β) r-1 The congurations in ∆ Q (α, β) for some partition Q and 0 < α < β are made of `well-separated' clusters.

Moment of order 4

The moment of order 4 plays a special role in the proof of tightness.

Let J = [b, b + k] be an interval. We will bound the moment of order 4 of Y = j∈J T Z j (ω) f , by using (43) and by bounding the cumulant of order 4: C (4) ( j∈J T Z j (ω) f ). We have:

|C (4) ( j∈J T Z j (ω) f )| ≤ H(ω, b, k) := s,t,v,w∈J |C(T Zs(ω) f, T Zt(ω) f, T Zv(ω) f, T Zw(ω) f )|.
Observe that H is super-additive in the sense of the denition given later in Section 5.

Partitions of {1, 2, 3, 4}

In Formula (41) of cumulants, the contribution of a partition such that one of its atoms is a singleton is 0, and so does not appear. The partitions of {1, 2, 3, 4} without atoms reduced to a singleton are {{1, 2, 3, 4}}, {{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}}.

The cumulant of order 4 of (T

h i f, T h k f, T h f, T hr f ) reads: C(T h i f, T h k f, T h f, T hr f ) = E(T h i f T h k f T h f T hr f ) (44)
-

[E(T h i f T h k f ) E(T h f T hr f ) + E(T h i f T h f ) E(T h k f T hr f ) + E(T h i f T hr f ) E(T h k f T h f )].

Well separated congurations

The following proposition is a key step in the proof of the CLT shown in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF].

Proposition 3.2. (Proposition 6.2 in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF]) For every sequence (β j ) j=0,...,r such that

β 0 = 0 < β 1 < 3β 1 < β 2 < ... < β r-1 < 3β r-1 < β r ,
we have

H r = ∆(β r ) ∪ r-1 j=0 |Q|≥2 ∆ Q (3β j , β j+1 ) . (45)
Below, as an illustration, we give a proof for r = 4.

Proof. Given a set H = {t, u, v, w} with 4 elements in H, we will show that (up to a permutation) H belongs to one of the conguration types

S 0 = ••••, S 1 = ••• •, S 2 = •• ••, S 3 = •• • •, or S 4 = • • • •.
The congurations of type S 0 are the `clustered' congurations. This is quantied by saying that these congurations are in ∆(β) for some β > 0. The congurations of type S 4 are the congurations with pairwise distant elements. They correspond to ∆ Q (0, β) for some β.

We show that H is either of type S 0 , with d(x, y) < β

d(t, u) < β 1 , d(t, w) > β 4 ≥ β 3 , d(u, w) > β 4 -β 1 ≥ β 3 . (46)
One of the following cases occurs:

1) (type S 1 = • • • •) d(v, t) < 2β 2 , which implies: d(v, u) < 2β 2 + β 1 < 3β 2 , d(v, w) > β 4 -2β 2 > 3β 3 -2β 2 > β 3 .
In this case, for the partition Q = {{t, u, v}, {w}},

we get d Q (H) < 3β 2 and d Q (H) > β 3 . 2) d(v, t) > 2β 2 , which implies: d(v, u) > 2β 2 -β 1 > β 2 , 2a) (type S 2 = • • ••) d(v, w) < β 2 , which implies: d(v, t) > d(t, w) -d(v, w) > β 4 -β 2 > 3β 3 -β 2 > β 3 , d(v, t) > d(t, w) -d(v, w) > β 4 -β 1 -β 2 > 3β 3 -β 1 -β 2 > β 3 . For Q = {{t, u}, {v, w}}, we get d Q (H) < 3β 2 and d Q (H) > β 3 . 2b) (type S 3 = • • • •) d(v, w) > β 2 : For Q = {{t, u}, {v}, {w}}, we get d Q (H) < 3β 1 and d Q (H) > β 2 .

A sucient condition for the CLT.

Let us recall a criterium in terms of cumulants for the CLT (cf. [START_REF] Leonov | On the central limit theorem for ergodic endomorphisms of compact commutative groups[END_REF]Th. 7], [8, Th. 6.2]). It is convenient to formulate the criterium by using a summation sequence, w = (w n ) n≥1 , i.e., for each n a function w n :

∈ Z d → w n ( ) ∈ R, with 0 < ∈Z d |w n ( )| < +∞.
The associated normalized non-negative kernel is K(w n )(t) =

| ∈Z d w n ( ) e 2πi ,t | 2 ∈Z d |w n ( )| 2 , t ∈ T d .
As for the summation along a random walk which is a special case, we say that the summation is ξ-regular for a probability measure ξ on T d , if the normalised kernel ( K(w n ) n≥1 ) converges weakly to ξ, i.e.,

lim n→∞ T d K(w n ) ϕ dt = ξ(ϕ), for every continuous function ϕ on T d . ( 47
)
This implies that for f , under Condition (5), the asymptotic variance for the normalised sums is

σ 2 w (f ) := lim n ∈Z d w n ( ) T f 2 2 ∈Z d |w n ( )| 2 = ξ(ϕ f ).
Theorem 3.3. If (w n ) n≥1 is a summation sequence on Z d such that (47) holds for a probability measure ξ on T d , the condition

( 1 ,..., r ) ∈(Z d ) r w n ( 1 )...w n ( r ) C(T 1 f, ..., T r f ) = o ( ∈Z d w 2 n ( )) r 2 , ∀r ≥ 3, (48)
implies

∈Z d w 2 n ( ) -1 2 ∈Z d w n ( )f (T .) =⇒ n→∞ N (0, ξ(ϕ f )). (49)
The following result (cf. [8, Lemma 6.6]) shows that mixing of all orders implies the asymptotic nullity of the cumulants.

Proposition 3.4. Let (T , ∈ Z d ) be a Z d -measure preserving action on a probability space (E, µ). If it is mixing of order r ≥ 2, then, for any f ∈ L ∞ 0 (X), lim

max i =j i -j →∞ C(T 1 f, ..., T r f ) = 0. (50)
Remark that (50) does not give the quantitative estimate (48). Nevertheless, in Section 6, (50) will be sucient for an action by automorphisms of a connected compact abelian group (in particular of a torus) which is mixing, when f is a trigonometric polynomial. For general exponentially mixing actions, a quantitative formulation is needed as in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF], using Proposition 3.2.

Array of sequences and nite dimensional distributions

For s ≥ 1 and j = 1, ..., s, let (w n,j , n ≥ 1) be s summation sequences, satisfying (47) with respectively ξ = ξ j , where the ξ j 's are probability measures on T d .

Using Theorem 3.3, we are going to deduce from the following two conditions the asymptotic normality (after normalization) of the vectorial process

∈Z d w n,1 ( ) T f, ..., ∈Z d w n,s ( ) T f :
-asymptotic orthogonality:

T d ( ∈Z d
w n,j ( ) e 2πi ,t ) (

∈Z d
w n,j ( ) e -2πi ,t ) e -2πi p,t dt

= o ∈Z d |w n,j ( )| 2 + ∈Z d |w n,j ( )| 2 , ∀j = j , ∀p ∈ Z d . (51)
-convergence to 0 of the normalized cumulants of order ≥ 3:

( 1 ,..., r ) ∈(Z d ) r w n,i 1 ( 1 )...w n,ir ( r ) C(T 1 f, ..., T r f ) = o( ∈Z d [ s j=1
|w n,j ( )| 2 ]) r/2 , ∀(i 1 , ..., i r ) ∈ {1, ..., s} r , ∀r ≥ 3.

(52) Proposition 3.5. Under Conditions (51) and ( 52), the vectorial process

∈Z d w n,1 ( ) T f ( ∈Z d |w n,1 ( )| 2 ) 1 2 , ..., ∈Z d w n,s ( ) T f ( ∈Z d |w n,s ( )| 2 ) 1 2 n≥1
is asymptotically distributed as N (0, J s ), where J s is the s-dimensional diagonal matrix with diagonal (ξ j (ϕ f ), j = 1, ..., s).

Proof. The hypothesis (51) implies, for s non zero real parameters a 1 , ..., a s :

( j a 2 j ∈Z d |w n,j ( )| 2 ) -1 | ∈Z d j a j w n,j ( ) e 2πi ,t | 2 weakly -→ n→∞ ( j a 2 j ) -1 j a 2 j ξ j . ( 53
)
Putting w a 1 ,...,as n ( ) = a 1 w n,1 ( ) + ... + a s w n,s ( ), by the Cramér-Wold theorem, to conclude it suces to show

∈Z d w a 1 ,...,as n ( )T f (a 2 1 ∈Z d |w n,1 ( )| 2 + ... + a 2 s ∈Z d |w n,s ( )| 2 ) 1 2 =⇒ N (0, s j=1 a 2 j ξ j (ϕ f )/ s j=1 a 2 i ). (54)
By ( 52), the sum i 1 ,...,ir∈{1,...,s} r ( 1 ,..., r ) ∈(Z d ) r

w n,i 1 ( 1 )...w n,ir ( r ) C(T 1 f, ..., T r f ) satises ( 48)

and the result follows from Theorem 3.3.

We will use also the following lemma. Let (w n ) n≥1 be a summation sequence on Z d such that (47) holds for a probability measure ξ on T d . For f ∈ L 2 (µ), we put σ n (f ) := w n ( ) T f 2 . We can suppose ξ(ϕ f ) > 0, since otherwise the limiting distribution is δ 0 . Lemma 3.6. Let f, f k , k ≥ 1 be in L 2 (µ) and satisfying [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF]

such that ϕ f -f k ∞ → 0. Then ∈Z d w n ( ) T f k σ n (f k ) =⇒ n→∞ N (0, 1), ∀k ≥ 1, implies ∈Z d w n ( ) T f σ n (f ) =⇒ n→∞ N (0, 1).
Proof. Let us consider the processes dened respectively by

U k n := ( ∈Z d w 2 n ( )) -1 2 ∈Z d w n ( ) T f k , U n := ( ∈Z d w 2 n ( )) -1 2 ∈Z d w n ( ) T f.
By (47) we have:

( ∈Z d w 2 n ( )) -1 ∈Z d w n ( ) T f 2 2 = T d K(w n ) ϕ f dt → n→∞ ξ(ϕ f ).
Observe that, if ξ is a probability measure on T d , f → (ξ(ϕ f ))

1 2 satises the triangular inequal- ity; hence ξ(ϕ f k ) ≥ ξ(ϕ f ) -ξ(ϕ f -f k ). Since ξ(ϕ f -f k ) ≤ ϕ f -f k ∞ → 0, this implies ξ(ϕ f k ) = 0 for k big enough. The hypotheses imply U k n =⇒ n→∞ N (0, ξ(ϕ f k )) for every k. Moreover, since lim n |U k n -U n | 2 2 dµ = lim n T d K(w n ) ϕ f -f k dt = ξ(ϕ f -f k ) ≤ ϕ f -f k ∞ , we have lim sup n µ[|U k n -U n | > δ] ≤ δ -2 lim sup n |U k n -U n | 2 2 dµ → k→∞ 0, for every δ > 0.
Therefore, using [2, Theorem 3.2], the conclusion U n =⇒ n→∞ N (0, ξ(ϕ f )) follows.

Moving averages of i.i.d. random variables

Let (X ) ∈Z 2 be a r.f. of centered i.i.d. real random variables on a probability space (E, B, µ) such that X 0 2 = 1. Let (a q ) q∈Z 2 be an array of real numbers such that q∈Z 2 |a q | < ∞ and let Ξ = (Ξ ) ∈Z 2 be the random eld dened by Ξ (x) = q∈Z 2 a q X -q (x).

The correlation is ϕ Ξ ( ) = q∈Z 2 a q X -q , q ∈Z 2 a q X -q = q∈Z 2 a q a q-. We have

| ϕ Ξ ( )| ≤ q∈Z 2 |a q | |a q-| = ( q∈Z 2 |a q |) 2 < +∞.
The continuous spectral density of the process is ϕ Ξ (t) = | q a q e 2πi q,t | 2 . The asymptotic variance for the summation along the r.w. (with the normalisation by C 0 n log n) is ( q∈Z 2 a q ) 2 that we suppose = 0.

Using the method of associated r.v.s we obtain a quenched FCLT for S ω,Ξ nt (cf. Notation (1)):

Theorem 4.1. Let (Ξ , ∈ Z 2 ) be the random eld dened by moving averages as above. Under the assumption q∈Z 2 |a q | < ∞, the process

S ω,Ξ nt (x) √ n log n t≥0 satises a quenched FCLT with asymptotic variance σ 2 = | q∈Z 2 a q | 2 (π √ det Σ) -1 .
Proof. 1) Convergence of the nite dimensional distributions a) First we assume that the random variables X , ∈ Z 2 , are bounded. Moreover let us consider rst a nite sum F = s∈S a s X s , where S is a nite subset of Z 2 . The case of the series, Ξ 0 = s∈Z 2 a s X s , will follow by an approximation argument.

As we have seen, for Y n (ω,

x, t) = S ω,X [nt] (x) √ C 0 n log n , we have to show: ∀ 0 = t 0 < t 1 < ... < t r = 1, (Y n (t 1 ), ..., Y n (t r )) =⇒ n→∞ (W t 1 , ..., W tr ).
For it, we use Proposition 3.5. Condition (51) follows from Lemma 1.10. Let us check (52).

There is M such that the cumulant C(T 1 F, ..., T r F ) = 0, if max i,j ij > M , because if M is big enough, there is a random variable T i F which is independent from σ-algebra generated by the others in the collection T 1 F, ..., T r F (by niteness of S).

| max i,j i -j ≤M C(T 1 F, T 2 F, ..., T r F )| w i 1 n (ω, 1 ) w i 2 n (ω, 2 )...w ir n (ω, r ) ≤ j 2 ,..., j r ≤M, j 1 =0 |C(T F, T +j 2 F, ..., T +j r F )| r k=1 w i k n (ω, + j k ) ≤ C j 2 ,..., j r ≤M, j 1 =0 r k=1 w i k n (ω, + j k ) ≤ C j 2 ,..., j r ≤M, j 1 =0 r k=1 w n (ω, + j k ).
The right hand side is less than a nite sum of sums of the form ∈Z d r k=1 w n (ω, + j k ) with {j 1 , ..., j r } ∈ Z 2 . By [START_REF] Móricz | Moment inequalities and the strong laws of large numbers[END_REF], for every ε > 0, there is C ε (ω) a.e. nite such that sup w n (ω, ) ≤ C ε (ω) n ε . For r ≥ 3, take ε < r-2

2(r-1)

. We have then

∈Z d r k=1 w n (ω, + j k ) ≤ C ε (ω) r-1 n ε(r-1) n = o(n r/2 )
and ( 52) is satised.

Using Lemma 3.6, the result can be extended to a general sum s∈S a s X s , such that s∈S |a s | < ∞.

b) Now if we assume only the condition X 0 2 < ∞, we apply a truncation argument to the independent r.v.s X and use again Lemma 3.6.

2) Tightness Let a + q = max(a q , 0), a - q = max(-a q , 0). Observe that the random variables q∈Z 2 a + q X -q (x) = q∈Z 2 a + q+ X -q (x), for ∈ Z 2 , are associated (as non decreasing functions of associated r.v.s), as well as q∈Z 2 a - q X -q (x), for ∈ Z 2 .

Therefore tightness can be proved separately for both processes. Now, again we use a truncation of the independent r.v.s X . For L > 0, we put

XL k := X k 1 {|X k |≤L} -E(X k 1 {|X k |≤L} ), XL k := X k -XL k = X k 1 {|X k |>L} -E(X k 1 {|X k |>L} ).
The r.v.s ( q∈Z 2 a + q+ X-q (x), ∈ Z 2 ) are still associated, as well as ( q∈Z 2 a + q+ X-q (x), ∈ Z 2 ).

Moreover, we control the norms in L 2 (µ) and L 4 (µ), since by the triangular inequality, it holds:

B j=A q∈Z 2 a + q XZ j (ω)-q µ,4 ≤ ( q∈Z 2 a + q ) B j=A XZ j (ω) µ,4 , B j=A q∈Z 2 a + q XZ j (ω)-q µ,2 ≤ ( q∈Z 2 a + q ) B j=A XZ j (ω) µ,2 .
Now the proof is like the proof of tightness in Theorem 2.2.

Tightness and 4th-moment

In this section, we show a criterium of tightness based on the 4th-moment.

Let n ≥ 1. We say that a nonnegative function

G 0 = (G 0 (b, k)), dened for b, k such that 0 ≤ b ≤ b + k ≤ n, is super-additive if G 0 (b, 0) = 0 and G 0 (b, k) + G 0 (b + k, ) ≤ G 0 (b, k + ), ∀b ≥ 0, ∀k, ≥ 1 such that b + k + ≤ n. (55)
Let (W k ) be a sequence of real or complex random variables on a probability space (E, µ). We set

S b,k = b+k r=b+1 W r , M b,n = max 1≤k≤n |S b,k |.
The following result is adapted from [START_REF] Móricz | Moment inequalities and the strong laws of large numbers[END_REF]:

Theorem 5.1. (F. Móricz) Let n ≥ 1. Suppose that there exists G 0 a super-additive function such that

E µ (|S b,k | 4 ) ≤ G 2 0 (b, k), ∀ b, k such that 0 ≤ b ≤ b + k ≤ n. (56)
Then, with the constant

C max = (1 -2 -1 4 ) -4 , E µ (|M b,n | 4 ) ≤ C max G 2 0 (b, n), ∀b ≤ n. (57)
Let X = (X ) ∈Z 2 be a strictly stationary real random eld on a probability space (E, µ), where the X 's have zero mean and nite second moment. Setting S ω J (x) = i∈J X Z i (ω) (x) if J is an interval, we deduce from (57) a criterium for tightness adapted to the sums along a random walk.

Proposition 5.2. Let G(ω, ., .), H(ω, ., .) be super-additive functions such that for a parameter γ and K 1 (ω), K 2 (ω) a.e. nite functions on (Ω, P),

G(ω, b, k) ≤ K 1 (θ b ω) k log k, H(ω, b, k) ≤ K 2 (θ b ω) k (log k) γ , G(ω, b, k) ≥ k. (58)
Suppose that the r.v.s X are bounded and satisfy

E µ (|S ω,X J | 4 ) ≤ G(ω, b, k) 2 + n 1 2 (log n) -(γ+1) H(ω, b, k), ∀J = [b, b + k] ⊂ [1, n], for a.e. ω. (59)
Then, for every

ε > 0, Y n (ω, x, t) = 1 √ n log n [nt] j=1 X Z j (ω) satises lim δ→0 + lim sup n µ(x ∈ E : sup |t -t|≤δ |Y n (ω, x, t ) -Y n (ω, x, t)| ≥ ε) = 0. (60) Proof. 1) Let c ≥ 0, ∆ n = [n 1 2 (log n) -2 ]+1, ν = ν n ≥ ∆ n , L n = [ νn ∆n ], ν = ν n = [ νn ∆n ]∆ n +∆ n -1.
The integer ν n will be chosen of order δn. Since ∆ n is an integer ≥ 1, we have ν ≤ ν . We can write, with the convention that -1

r=0 = 0: max 0≤k≤ν | k j=0 X Z j+c (ω) | ≤ max 0≤k≤ν | k j=0 X Z j+c (ω) | = max 0≤u≤[ ν ∆n ],1≤k≤∆n | u-1 r=0 (r+1)∆n-1 j=r∆n X Z j+c (ω) + u∆n+k-1 j=u∆n X Z j+c (ω) | ≤ max 0<u≤Ln, 1≤k≤∆n | u-1 r=0 (r+1)∆n-1 j=r∆n X Z j+c (ω) | + max 0≤u≤Ln, 1≤k≤∆n | u∆n+k-1 j=u∆n X Z j+c (ω) | = max 0<u≤Ln | u∆n j=0 X Z j+c (ω) | + max 0≤u≤Ln, 1≤k≤∆n | u∆n+k-1 j=u∆n X Z j+c (ω) |.
Let Ân and Ãn be respectively dened as the rst and the second term above. This previous inequality implies

µ( max 0≤k≤ν | k j=0 X Z j+c (ω) | ≥ ε n log n) ≤ µ( Ân ≥ 1 2 ε n log n) + µ( Ãn ≥ 1 2 ε n log n). (61)
For Ãn , since the X 's are bounded (uniformly in by stationarity), by the choice of

∆ n there is N 1 (ε, δ) such that µ( Ãn ≥ 1 2 ε √ n log n) = 0, for n ≥ N 1 (ε, δ).
For Ân we will apply Theorem 5.1 to W r = (r+1)∆n-1 j=r∆n

X Z j+c (ω) , with G 0 (b, k) := G(ω, c + b∆ n , k∆ n ) + (log n) -γ+1 H(ω, c + b∆ n , k∆ n ), (62)
which is super-additive as G and H.

Since G(ω, c + b∆ n , k∆ n ) ≥ k∆ n ≥ ∆ n = [n 1 2 (log n) -2 ],
we have for k ≥ 1:

G 2 (ω, c + b∆ n , k∆ n ) + n 1 2 (log n) -γ-1 H(ω, c + b∆ n , k∆ n ) ≤ G 2 0 (b, k).
Therefore, by (59),

E µ (| b+k r=b+1 W r | 4 ) = E µ (| (b+k+1)∆n-1 j=(b+1)∆n X Z j+c (ω) | 4 ) ≤ G 0 (b, k) 2 , ∀b ≥ 0, ∀k ≥ 1,
which implies by (57) of Theorem 5.1:

E µ ( max 1≤k≤p | (b+k+1)∆n-1 j=(b+1)∆n X Z j+c (ω) | 4 ) ≤ C max G 0 (b, p) 2 , ∀b ≥ 0, ∀p ≥ 1.
Putting K(ω) := max(K 1 (ω), K 2 (ω)) and using (58), we get the bound

Ln max u=1 | u∆n j=0 X Z j+c (ω) | 4 4 ≤ C max [G(ω, c, L n ∆ n ) + (log n) -γ+1 H(ω, c, L n ∆ n )] 2 ≤ C max K(θ c ω) 2 [L n ∆ n log(L n ∆ n ) + (log n) -γ+1 (L n ∆ n )(log(L n ∆ n )) γ ] 2 . (63)
2) For M > 0 big enough, the set Ω M := {ω :

K(ω) ≤ M } has a probability P(Ω M ) ≥ 1 2 . We apply Lemma 1.7 to Ω M . Given δ > 0, there is N 2 (δ) such that for n ≥ N 2 (δ), we can nd a sequence 0 = ρ 1,n < ρ 2,n < ... < ρ v,n ≤ n < ρ v+1,n of visit times of θ k ω in Ω M under the iteration of the shift θ, such that 1 2 δn ≤ ρ i+1,n -ρ i,n ≤ 3 2 δn and v < 2/δ. By construction, K(θ ρ i,n ω) ≤ M, ∀i. With c = ρ i,n , ν n = ν i,n = ρ i+1,n -ρ i,n ≤ 3 2 δn, L i,n = [ ν i,n
∆n ] (so that L i,n ∆ n ≤ 3 2 δn), we deduce from the upper bound (63) (for n big enough and using 0 ≤ log(δn) ≤ log n, if n ≥ δ -1 ):

L i,n max u=1 | u∆n j=0 X Z ρ i,n +j (ω) | 4 4 ≤ C max M [ν i,n log ν i,n + (log n) -γ+1 ν i,n (log ν i,n ) γ ] 2 ≤ C max M [ 3 2 δn log(δn) + (log n) -γ+1 3 2 δn(log(δn)) γ ] 2 ≤ C max M [3δn log n] 2 .
This implies that there is a constant C such that, for n ≥ N 2 (δ), for i = 1, ..., v: µ( max

0≤u≤L i,n | u∆n-1 j=0 X Z j+ρ i (ω) | ≥ 1 2 ε n log n) ≤ C max M (3δn log n) 2 ( 1 2 ε √ n log n) 4 ≤ C ε -4 δ 2 .
This allows to bound the term µ( Ân

≥ 1 2 ε √ n log n) in (61): Putting t i = ρ i,n /n, we obtain, for n ≥ max(N 1 (ε, δ), N 2 (δ)), µ( sup |t -t|≤δ |Y n (t ) -Y n (t)| ≥ 3ε) ≤ v i=1 µ( sup t i-1 ≤s≤t i |Y n (s) -Y n (t i-1 )| ≥ ε) ≤ 2C ε -4 δ 2 v ≤ 2C δ ε 4 .
Remark 5.3. Let be given for each s in a set of indices S a process X s = (X s, ) ∈Z 2 satisfying the hypotheses of the proposition, with the same uniform bound and the same G, H, γ. Then, if X = s a s X s s, with s |a s | ≤ 1, the r.f. X = (X ) satises the conditions of the proposition and therefore the conclusion (60). This follows from Minkowski inequality:

S ω,X J 4 4 ≤ ( s |a s | S ω,X s J 4 ) 4 ≤ ( s |a s |[G(ω, b, k) 2 + n 1 2 (log n) -(γ+1) H(ω, b, k)] 1 4 ) 4 = ( s |a s |) 4 [G(ω, b, k) 2 + n 1 2 (log n) -(γ+1) H(ω, b, k)].

Automorphisms of a torus

We consider now a random eld generated by the action of commuting automorphisms on a torus. Let us rst present the model. We will give the details of the proof for d = 2.

Actions by endomorphisms on a compact abelian group: Let G be a compact abelian group with Haar measure µ. The group of characters of G is denoted by Ĝ and the set of non trivial characters by Ĝ * . The Fourier coecients of a function

f in L 1 (G, µ) (denoted also f (k) when G is a torus) are c f (χ) := G χ f dµ, χ ∈ Ĝ.
Every surjective endomorphism of G denes a measure preserving transformation on (G, µ) and a dual injective endomorphism on Ĝ. Let AC 0 (G) denote the space of real functions on G with absolutely convergent Fourier series and µ(f ) = 0, endowed with the norm:

f c := χ∈ Ĝ |c f (χ)| < +∞.
Recall that the action on G is mixing of all orders when it is totally ergodic and G is connected.

Proposition 6.1. If f is in AC 0 (G), the spectral density ϕ f is continuous on T d and ϕ f ∞ ≤ f 2 c .
For every ε > 0 there is a trigonometric polynomial P such that ϕ f -P ∞ ≤ ε. Proof. Since by total ergodicity the characters T χ for ∈ Z d are pairwise distinct, we have

∈Z d | T f, f | ≤ ∈Z d χ∈ Ĝ * |c f (T χ)| |c f (χ)| ≤ χ∈ Ĝ * ( ∈Z d |c f (T χ)|) |c f (χ)| ≤ ( χ∈ Ĝ * |c f (χ)|) 2 . Therefore, if f is in AC 0 (G), then ∈Z d | T f, f | < ∞
, the spectral density is continuous and ϕ f ∞ ≤ ε. By this inequality, we can take for P the restriction of the Fourier series of f to a nite set E in Ĝ, where E is such that ϕ f -P ∞ ≤ ( χ∈ Ĝ\E |c f (χ)|) 2 ≤ ε.

For compact abelian groups which are connected (cf. [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]) or which belong to a special family of non connected groups (cf. [START_REF] Cohen | Almost mixing of all orders and CLT for some Z d -actions on subgroups of F Z d p[END_REF]), a CLT has been shown for summation either over sets or along a random walk. Our aim is to extend this latter result to a functional CLT at least in the case of automorphisms of a torus.

Matrices and automorphisms of a torus:

Now we will restrict to the special case of G = T ρ , ρ > 1.

Every A in the semigroup M * (ρ, Z) of non singular ρ × ρ matrices with coecients in Z denes a surjective endomorphism of T ρ and a measure preserving transformation on (T ρ , µ). It denes also a dual endomorphism of the group of characters T ρ identied with Z ρ (this is the action by the transposed matrix, but since we compose commuting matrices, for simplicity we do not write the transposition). The linear operator on C ρ dened by A is denoted by Ã.

When A is in the group GL(ρ, Z) of matrices with coecients in Z and determinant ±1, it denes an automorphism of T ρ . Recall that the action of A ∈ M * (ρ, Z) on (T ρ , µ) is ergodic if and only if A has no eigenvalue root of unity.

Here we present the proof for the case of automorphisms and for d = 2 (in the recurrent case for the random walk). Let (A 1 , A 2 ) be two commuting matrices in GL(ρ, Z) and A = A 1 1 A 2

2 , for = ( 1 , 2 ) ∈ Z 2 . It denes a Z 2 -action (A , ∈ Z 2 ) on (T ρ , µ), which is totally ergodic if and only if A has no eigenvalue root of unity for = 0.

Explicit totally ergodic Z 2 -actions can be computed (cf. [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]) like the example below (see the book of H. Cohen on computational algebraic number theory [START_REF] Cohen | A course in computational algebraic number theory[END_REF]):

A 1 =   -3 -3 1 10 9 -3 -30 -26 9   , A 2 =   11 1 -1 -10 -1 1 10 2 -1   .
We will need an algebraic result based on the following theorem on S-unit equations ( [START_REF] Schlickewei | S-unit equations over number elds[END_REF]): Theorem 6.2. ([17, Th. 1.1]) Let K be an algebraically closed eld of characteristic 0 and for r ≥ 2, let Γ r be a subgroup of the multiplicative group (K * ) r of nite rank. For any (a 1 , ..., a r ) ∈ (K * ) r , the number of solutions x = (x 1 , ..., x r ) ∈ Γ r of the equation a 1 x 1 + ... + a r x r = 1 such that no proper subsum of a 1 x 1 + ... + a r x r vanishes, is nite.

Corollary 6.3. Suppose that the Z 2 -action (A , ∈ Z 2 ) is totally ergodic. The set F of triples

( 1 , 2 , 3 ) ∈ (Z 2 ) 3 for which there is γ ∈ Z 2 \ {0} such that A 1 γ -A 2 γ + A 3 γ -γ = 0, (64)
without vanishing proper sub-sum, is nite. Proof. There exists a decomposition of E = C ρ into vectorial subspaces C ρ = ⊕ k E k which are simultaneously invariant by Ãi , i = 1, 2, and such that there is a basis B k in which Ãi restricted to E k is represented in a triangular form with an eigenvalue of Ãi on the diagonal.

This follows from the fact that the commuting matrices A i have a common non trivial space W of eigenvectors, and then by an induction on the dimension of the vector space, applying the induction hypothesis to the action of the quotient map of Ãi on E/W .

For γ ∈ Z ρ \ {0}, there is k 0 such that the component γ 0 of γ in E k 0 is = 0. Let δ 0 be the dimension of E k 0 . In the basis B k 0 = {e k 0 ,1 , ..., e k 0 ,δ 0 } of E k 0 , we denote the coordinates of γ 0 by (γ 1 0 , ..., γ δ 0 0 ). There is δ 0 ∈ {1, ..., δ 0 } such that γ i 0 = 0, ∀i < δ 0 , and v 0 := γ δ 0 0 = 0.

Due to the triangular form, for j = 1, 2, we have A j γ 0 = α k 0 ,j γ δ 0 0 e k 0 ,δ 0 + ζ(j, ), ∀ ∈ Z, where α k 0 ,j is an eigenvalue of A j and where ζ(j, ) belongs to the subspace generated by {e k 0 ,δ 0 +1 , ..., e k 0 ,δ 0 }.

Using the notation α u = α 1 u,1 α 2 u,2 , if α u,1 (resp. α u,2 ) is an eigenvalue of A 1 (resp. A 2 ), if (64) holds, then (α 1 k 0 -α 2 k 0 + α 3 k 0 )v 0 = v 0 .
This equation is still without vanishing proper sub-sum, because of the assumption of total ergodicity. By Theorem 6.2 applied to the multiplicative group (of nite rank) generated by α k 0 ,j , j = 1, 2, the number of solutions of the previous equation is nite. Hence the result, since k 0 takes a nite number of values.

Random walks and quenched CLT

Our aim is to replace the r.f. of i.i.d. variables (X , ∈ Z 2 ) discussed in Section 2 by the random eld generated by an observable f on a torus T ρ under the action of commuting automorphisms.

More precisely, we consider → A a totally ergodic Z 2 -action by algebraic automorphisms of T ρ , ρ > 1, dened by commuting ρ × ρ matrices A 1 , A 2 with integer entries, determinant ±1 such that the eigenvalues of A = A 1 1 A 2 2 are = 1, if = ( 1 , 2 ) = (0, 0). The composition with a function f dened on T ρ is denoted by A f as well as T f . We consider the random eld (X = A f, ∈ Z 2 ), with f ∈ AC 0 (T ρ ).

A sucient condition for f with 0 integral to be in AC

0 (T ρ ) is | f (k)| = O( k -β ), with β > ρ.
For a.e. ω the following asymptotic variance exists:

lim n 1 n log n n k=1 A Z k (ω) f 2 2 = C 0 k∈Z d T k f f ,
where the constant C 0 is dened in Subsection 1.2.

The following quenched FCLT extends for the torus the CLT proved in [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]. Remark that the CLT is proved therein for a general compact abelian group. The extension to a functional version of the CLT holds in this general case when f is a trigonometric polynomial.

Theorem 6.4. Let (Z n ) be a 2-dimensional reduced centered random walk with a nite moment of order 2 and let f be a real function in AC 0 (T ρ ) with spectral density ϕ f and ϕ f (0) = 0.

Denoting by S ω n (f ) := n k=1 A Z k (ω) f the sums along the r.w., the process

1 √ n log n S ω nt (f ) t∈[0,1]
satises the FCLT for a.e. ω.

Proof. 1) Convergence of the nite dimensional distributions 1a) First suppose that f is a trigonometric polynomial: f = k∈Λ c k (f ) χ k , where (χ k , k ∈ Λ) is a nite set of characters on T ρ and χ 0 the trivial character.

We use Proposition 3.5: (51) follows from [START_REF] Schlickewei | S-unit equations over number elds[END_REF] and Lemma 1.10. For (52), we have to show

C (r) ( ∈Z 2 w n (ω, ) T f ) = o((n log n) r/2 ), ∀r ≥ 3, for a.e. ω. (65)
We apply Theorem 3.3. Let us check (48). For r xed, the function (n 1 , ..., n r ) → m f (n 1 , ..., n r ) := X T n 1 f • • • T n r f dµ takes a nite number of values, since m f is a sum with coecients 0 or 1 of the products c k 1 ...c kr with k j in a nite set. The cumulants of a given order take also a nite number of values according to (42).

Therefore, since mixing of all orders implies lim max i,j ij →∞ C(T 1 f, ..., T r f ) = 0 by Proposition 3.4, there is M r such that C(T 1 f, ..., T r f ) = 0 if max i,j ij > M r . The end of the proof is then like in Theorem 4.1. 1b) For f ∈ AC 0 (T ρ ), using Proposition 6.1 and Lemma 3.6, the convergence follows by approximation of f by a sequence of trigonometric polynomials f L in such a way that lim L ϕ f -f L = 0.

2) Moment of order 4 and tightness

We use Proposition 5.2. Taking into account Remark 5.3, it suces for the tightness to take for f a character and show that the bounds are independent of the character.

Let χ v be a character on the torus T ρ , χ v : x → exp(2πi v, x ), where v ∈ Z ρ \ {0}.

For an interval J = [b, b + k] ⊂ [1, n], we have: b+k i=b A Z i χ v 4 4 = #{(i 1 , i 2 , i 3 , i 4 ) ∈ J 4 : (A Z i 1 -A Z i 2 + A Z i 3 -A Z i 4 ) v = 0}. This number is less than 2G 2 (ω, b, k) + H(ω, b, k), with G(ω, b, k) := #{(i 1 , i 2 ) ∈ J 2 : (A Z i 1 -A Z i 2 )v = 0}, H(ω, b, k) := #{(i 1 , i 2 , i 3 , i 4 ) ∈ J 4 : (A Z i 1 -A Z i 2 + A Z i 3 -A Z i 4 )v = 0},
where above in H we count the number of solutions without vanishing proper sub-sums.

By assumption of total ergodicity, if

(A Z i 1 -A Z i 2 )v = 0, then Z i 1 = Z i 2 , so that G is the number of self-intersections of the r.w. on [b, b + k]: G(ω, b, k) = V (ω, [b, b + k[) = V k (θ b ω). A bound of
these quantities is given by [START_REF] Leonov | On the central limit theorem for ergodic endomorphisms of compact commutative groups[END_REF].

For H, by Corollary 6.3, there is a nite set F (independent of v) such that

H(ω, b, k) ≤ #{(i 1 , i 2 , i 3 , i 4 ) ∈ J 4 : Z i 1 (ω) -Z i 4 (ω), Z i 2 (ω) -Z i 4 (ω), Z i 3 (ω) -Z i 4 (ω) ∈ F }.
Therefore, with the notation ( 24), H(ω, b, k)

≤ ( 1 , 2 , 3 )∈F W k (θ b ω, 1 , 2 , 3 ).
By [START_REF] Pène | Self-intersections of trajectories of the Lorentz process[END_REF] in Lemma 1.5, there exists a positive integrable function the asymptotic variance for the sums along the r.w. is dierent from 0 (cf. Subsection 1.2).

C 3 such that W n (ω, 1 , 2 , 3 ) ≤ C 3 (ω) n (log n) 5 , ∀n ≥ 1, which implies H(ω, b, k) ≤ (Card F ) C 3 (θ b ω) k (log k)
2) For automorphisms of a torus T ρ , in the recurrent 2-dimensional model studied above, if f satises the regularity condition |c f (k)| = O( k -β ), with β > ρ , then the asymptotic variance is given by ϕ f (0) and it is null, if and only if f is a mixed coboundary: there are continuous functions u 1 , u 2 such that f = (I -A 1 )u 1 + (I -A 2 )u 2 (cf. [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]).

7. Exponential mixing of all orders 7.1. FCLT and exponential mixing of all orders.

Our last example is given by commuting translations on homogeneous spaces. It relies on recent results on the exponential mixing of all orders for ows on homogeneous spaces shown in [START_REF] Björklund | Quantitative multiple mixing[END_REF] and their application to the CLT in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF]. Closely following the latter reference, we recall rst the notion of exponential mixing of all orders for a group action by measure preserving transformations.

Exponential mixing of all orders

Let H be a group with a left invariant distance d. Let h → T h be a homomorphism of H in the group of measure preserving invertible transformations of a probability space (E, B, µ). We denote by A a sub-algebra in L ∞ (E, µ) which is H-invariant. Let N = (N s ) be a family of semi-norms on A, indexed by positive integers s.

Hypothesis 7.1. The following conditions are assumed to hold for all s > 1 and all f, g ∈ A with constant factors depending only on s (all denoted by C s ):

1) N s (f ) ≤ C s N s+1 (f ); 2) f L ∞ ≤ C s N s (f ); 3) N s (f g) ≤ C s N s+1 (f ) N s+1 (g); 4
) there exists ζ s > 0 such that N s (T h f ) ≤ C s e ζsd(h,e) N s (f ), ∀h ∈ H.

Let r ≥ 2 be an integer. For H = (h 1 , ..., h r ) ∈ H r , we set d r (H) := min i =j d(h i , h j ).

Denition 7.2. We say that the H-action on (E, µ) is exponentially mixing of order r, with respect to d and (A, N ), if there exist constants C r,s , δ r > 0 and an integer s r > 0 such that for all s ≥ s r and f

1 , ..., f r ∈ A, |µ( r i=1 T h i f i ) - r i=1 µ(f i )| ≤ C r,s e -δrdr(H) r i=1 N s (f i ), (66)
for all H = (h 1 , ..., h r ) ∈ H r . The constant C depends only on r and s.

We may assume that ζ s is increasing with s, δ r decreasing with r and δ r < rζ s , ∀r, s.

In

Spectral density and cumulants

Let f be a centered function in A. Its spectral density is ϕ f = ∈Z d a e 2πi , t , with a = T f, f . The absolute summability |a | < ∞ is a consequence of (66) for r = 2.

Moreover, there exists s 2 and a constant

C such that ϕ f ∞ ≤ CN s 2 (f ) 2 . Let J = [b, b + k] ⊂ [1, n]. With the notation of 1.2.2, we have | j∈J T Z j (ω) f | 2 dµ = T d | j∈J e 2πi Z j (ω), t | 2 ϕ f (t) dt ≤ ∈Z d |a | V (ω, J, ) =: G(ω, b, k).
For a xed ω, the bound G(ω, b, k) is super-additive. By (43), we have

E µ (| j∈J T Z j (ω) f | 4 ) = 3 ( j∈J T Z j (ω) f ) 2 dµ 2 + C (4) ( j∈J T Z j (ω) f ). (67)
h ∈ H:

ν ω n (B(h, cγ)) = #{j ≤ n : Z j (ω) ∈ B(h, cγ)} ≤ sup w n (ω, ) Card(B(h, cγ)) ≤ K(ω)(cγ) 2 log 2 n. Due to Hypothesis 7.3 and by (69), this implies that there is a constant C 1 (depending only on r and s) such that |C (r) (ν ω n * f )| ≤ C 1 K(ω) r-1 (log n) 2(r-1) (cγ) 2(r-1) n + e -δrγ n r N s (f ) r . Taking γ = r-1 δr log n, the rst term above gives the bound (c r-1 δr ) 2(r-1) (log n) 4(r-1) n and the second term is n. It follows, for a constant C 2 depending on r and s: 1) n. Likewise for an interval J = [b, b + k] and ν ω J = j∈J δ Z j (ω) , we get |C (r) (ν ω J * f )| ≤ C 2 N s (f ) r K(θ b ω) r-1 (log k) 4(r-1) k.

|C (r) (ν ω n * f )| ≤ C 2 N s (f ) r K(ω) r-1 (log n) 4(r-
So we get the same type of upper bound for the cumulants as for the automorphisms of the torus. Therefore we have convergence of the nite dimensional distributions and tightness.

For a Z 2 -action satisfying the exponential mixing condition on an algebra of functions as presented at the beginning of this section, we can state now a functional version of a CLT result for the summation along a random walk. satises the FCLT for a.e. ω.

Remarks 7.7. 1) As suggested in a referee's remark, the above result can be extended to a space of functions larger than A. For r = 4, by Proposition 7.5, there is s 4 such that (69) holds for g ∈ A. Let A 1 be the space obtained as closure of A for the norm N s 4 .

Let us sketch a proof of the extension of Theorem 7.6 to functions f ∈ A 1 . By 7.2 we still have fast decorrelation for f ∈ A 1 . We can use Lemma 3.6 for f ∈ A 1 and (g n ) a sequence in A such that f -g n Ns 4 → 0 (hence also ϕ f -gn ∞ → 0 by (66) and hypothesis 7.1). Therefore f in A 1 satises the convergence of the nite dimensional distributions, as the functions g n in A. Moreover, by (69) the estimate for the cumulant of order 4, which is satised by functions in A, still holds for A 1 and this, as above, implies tightness.

2) The result is formulated for Z 2 , but an analogous result with a normalisation in √ n can be proved for a transient random walk in dimension ≥ 3 with the same method. To do it, we use the bound (67) for the 4th moment. In the transient case, the rst term behaves like n 2 because of [START_REF] Dembo | Thick points for planar Brownian motion and the Erdös-Taylor conjecture on random walk[END_REF]. The second term behaves as n(log n) 3(r-1) by using Prop. 7.3 and Theorem 13 in Erdos-Taylor [START_REF] Erdös | Some problems concerning the structure of random walk paths[END_REF] (which can be extended to a r.w. as for the upper bound in [START_REF] Dembo | Thick points for planar Brownian motion and the Erdös-Taylor conjecture on random walk[END_REF], according to Remark 1.4). Observe also that Proposition 5.2 can be adapted to the normalisation corresponding to the transient case.

Translations on homogeneous spaces.

The following example is an action which is exponentially mixing of all orders on an algebra A of functions according to [START_REF] Björklund | Quantitative multiple mixing[END_REF].

We take the group G = SL(n, R) and a lattice Γ in G, i.e., a discrete subgroup such that G/Γ has a nite volume for the measure µ induced by the Haar measure of G, for example SL(n, Z). The space E is the quotient G/Γ. The action on E will be given by left multiplication gΓ → hgΓ where h is in the diagonal subgroup D of G.

The algebra A in the example is the algebra of C ∞ -functions with compact support on G/Γ, and N is a family of Sobolev norms as in [START_REF] Björklund | Quantitative multiple mixing[END_REF].

Let us explicit the distance in the example.

Left invariant pseudo-metric on G = SL(n, R)

Recall that there is a distance on D, induced by a canonical left invariant distance on G. The left invariant distance d G dened on G is comparable on D to the `pseudo-metric' δ(., .) dened as follows:

For A ∈ G, let |||A||| = max x =0 Ax x be the norm of the n × n matrix A as operator on R n endowed with the euclidian norm. Since the determinant is 1, A has an eigenvalue of modulus ≥ 1, which implies |||A||| ≥ 1. For A, B ∈ G, we put δ(A, B) = log(|||A -1 B|||).

Clearly, δ(A, B) ≥ 0, the triangular inequality is satised by sub-multiplicativity of the operator norm, and δ is left invariant on G. If |||A||| = 1, the iterates of A are bounded, so all eigenvalues must have a modulus ≤ 1. As the determinant is 1, the modulus of the eigenvalues is 1. Now considering the Jordan form of A over C, it must be diagonal. Finally we conclude that {A : |||A||| = 1} is the orthogonal group in G.

We take two elements A 1 = exp(U 1 ), A 2 = exp(U 2 ), where U 1 , U 2 , in the sub-algebra D corresponding to D in the Lie algebra G of G, are such that U 1 , U 2 generate a 2-dimensional vectorial space of D. The group (A 1 1 A 2 2 , ( 1 , 2 ) ∈ Z 2 ) yields a totally ergodic action on G/Γ. In Theorem 7.6, in particular in the example provided by homogeneous spaces, as in the CLT in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF], the statement says nothing about the non-nullity of the variance, for a 2-dimensional recurrent r.w. The question of degeneracy of the asymptotic variance (for a recurrent r.w.) is the same as for the sums over squares: it depends of the nullity of ϕ(0). This contrasts with the action of commuting automorphisms of a torus, for which (as recalled in Remark 6.5.2) there is a description of the degenerate case in terms of mixed coboundaries.

In the case of a transient random walk on Z d , as noticed in Subsection 1.2, if the observable is non null a.e., the asymptotic variance for the sums along a transient r.w. is dierent from 0.

1 1 .

 1 ..T d d . The euclidean norm of ∈ Z d is denoted by | | or .

For a single

  random variable Y , the cumulant of order r is dened by C (r) (Y ) := C((Y, ..., Y ) r ), where (Y, ..., Y ) r is the vector with r components equal to Y . If Y is centered, we have Y 2 2 = C (2) (Y ) and

(r = 4 )

 4 We represent an element of H by • and, given two elements a, b ∈ H, draw •• or • • depending on whether they are close or far from each other.

1 Z

 1 j = , for u = 0, ..., r -1. Then we have w u n (ω, ) ≤ w n (ω, ) and, since sup 1 ,..., r |C(T 1 F, T 2 F, ..., T r F )| < ∞:

Let (T 1

 1 , ..., T d ) be a nite family of d commuting surjective endomorphisms of G and T = T 1 1 ...T d 1 , for = ( 1 , ..., d ) ∈ N d . We obtain a N d -action (T , ∈ Z d ) on G, which is totally ergodic if and only if the dual action is free.

Theorem 7 . 6 .

 76 Let (Z n ) be a 2-dimensional aperiodic centered random walk with moments of order 2. Let f be a real centered function in A with spectral density ϕ f such that ϕ f (0) = 0. Under Hypothesis 7.3, denoting by S ω n (f ) := n k=1 T Z k (ω) f the sums along the r.w., the process 1 √ n log n S ω nt (f ) t∈[0,1]

For instance in SL( 3 , 2 .h 2 2

 322 R), we can take with a 1 , a 2 > 1: A 1 = The measure ν n (B(Id, R)) is the counting measure with some weight applied to the ball, therefore up to the weight it is the number of elements of the form h 1 1 in the ball, and nally the (weighted) number of integers= ( 1 , 2 ) of norm ≤ R.By what precedes, Hypothesis 7.3 is satised and Theorem 7.6 yields a functional CLT in the class of centered compactly supported C ∞ -functions for the action of a 2-dimensional random walk on the diagonal subgroup on G/Γ. A result analogous to Theorem 7.6 holds for the sums along a transient random walk: the only change is the estimate of the number of self-intersection (normalisation by √ n).

  and otherwise transient. Recurrence occurs if and only if P(Z n = 0 innitely often) = 1, and transience if and only if

  4) The quenched FCLT shown in the dierent examples below is valid for a set of ω's of Pmeasure 1 given by the results of this Section 1. This set does not depend on the Z d -dynamical systems considered in the further sections. The joint distribution on Ω × E is used only when the annealed model is mentioned when Bolthausen's result is recalled in Section 2. (t), t ∈ [0, 1]) be a process on (E, µ) with values in the space C[0, 1] of real valued continuous functions on [0, 1] or in the space D[0, 1] of right continuous real valued functions on [0, 1] with left limits, endowed with the uniform norm.

	1.2.3. Formulation of the quenched FCLT.
	(For the reminders below, see Theorem 7.5, p. 84 in [2], as well as Theorems 15.1 and 15.5 in
	the rst edition of this book.)
	Let (Y n

Let (W (t), t ∈ [0, 1]) be the Wiener process on [0, 1]. To show a functional limit theorem (FCLT) for (Y n (t), t ∈ [0, 1]), i.e., weak convergence to the Wiener process, it suces to prove the two following properties ( =⇒ denotes the convergence in distribution):

  4 , for every x, y in H, or of type S 4 , with d(x, y) ≥ β 1 , for every x = y in H, or of one of the types S 1 , S 2 , S 3 . If H is not of type S 0 or S 4 , there are distinct elements, let call them t, u, w, such that d(t, u) < β 1 and d(t, w) > β 4 , which implies

  5 . Remark that the bounds do not depend on the character, but only on the matrices A 1 , A 2 .

Since G and H are super-additive (Condition (55)), the tightness property follows now from Proposition 5.2 with γ = 5. Remarks 6.5. 1) An analogous result is valid for any transient random walk in dimension d ≥ 1, with the standard normalisation by √ n. In this case, if the observable is non null a.e.,

  what follows, we will consider, for d ≥ 2, a measure preserving Z d -action on a probability space (E, B, µ) generated by d commuting invertible maps T 1 , ..., T d . Therefore the group H in Denition 7.2 is going to be the group Z d still denoted also by H. this assumption, for simplicity of notation we can assume that the distance d in Denition 7.2 applied to Z d is the Euclidean distance on Z d .

If (Z n ) is a random walk on Z d , then we get a random walk (T Zn(ω) ) on the group of measure preserving invertible transformations on (E, B, µ).

With a distance on Z d associated to a norm equivalent to the Euclidean norm, the volume of a big ball is of order the number of integral points in the ball. It is important to relate this distance to the distance d of Denition 7.2. We will assume that the action satises: Hypothesis 7.3. For an H-invariant sub-algebra A in L ∞ (E, µ) and a family N = (N s ) of semi-norms on A, we assume that the Z d -action on (E, B, µ) is exponentially mixing of order r for every r ≥ 2 in the sense of Denition 7.2 with a distance d equivalent to the Euclidean distance.

With

The next proposition shows that the cumulants C(T h 1 f, ..., T hr f ) are small for all well-separated r-tuples H = (h 1 , ..., h r ). The constants δ r and ζ r are those of Denition 7.2. Proposition 7.4. (Proposition 6.1 in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF]) For r ≥ 2, let Q be a partition of {1, 2, ..., r} with |Q| ≥ 2 and let s be an integer

where the constant C depends only on r and s.

Proof. (r = 4) Let us give the proof for r = 4 and in the case of the congurations

We may write the formula for the cumulants in the following way:

We use the exponential mixing of order 2 for (B), (C), (D) and of order 3 for (A). More precisely, we have

Therefore, by the 3-mixing applied to f T h k -h i f, T h -h i f, T hr-h i f , we have (A) ≤ C e 3ζs β j e -δr β j+1 N s+1 (f ) 2 N s (f ) 2 .

Let us now x γ > 0. We dene β j by β 0 = 0 and β j+1 = 3rβ j ζ s /δ r + γ, for j > 0.

As δ r < rζ s , we have β j+1 > 3β j . Moreover β r = γ r-1 j=0 (3r ζ s /δ r ) j =: γc r,s . Let ν = ν( )δ be a positive nite measure on H = Z d with mass ν . Let B(h, R) be the euclidean ball of center h and radius R. The following bound on cumulants is given in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF] (cf. Proposition 7.4 above and (45)). The rst term at right in (69) comes from the clustered congurations and the second term from the well separated congurations. The constants s r and δ r are introduced in Denition 7.2. Proposition 7.5. (Proposition 5.2 in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF]) For every r ≥ 3 and s ≥ s r + r, there exist C r,s and c r,s > 0 (not depending on ν) such that, for all γ > 0 and f ∈ A,

Under the assumption of exponential mixing of all orders, it is shown in [START_REF] Björklund | Central limit theorems for group actions which are exponentially mixing of all orders[END_REF] that the CLT holds for ν n * f , when f ∈ A and (ν n ) a sequence of measures satisfying a certain condition.

In our framework, ν is the measure ν ω n = n-1 j=0 δ Z j (ω) , where (Z j ) j≥0 is a r.w. on Z 2 . Its mass is ν ω n = n. The convolution ν * f in (69) for ν ω n means n-1 j=0 f (T Z j (ω) .). Let us assume that the r.w. Z is a centered random walk on Z 2 with moments of order 2. By (23) (cf. Remark 1.4 in 1.2.2), it holds with a constant K(ω) nite for a.e. ω, uniformly in