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

ON THE QUENCHED FUNCTIONAL CLT IN 2-D RANDOM SCENERIES, EXAMPLES

Keywords: 2010 Mathematics Subject Classication. Primary: 60F05, 28D05, 22D40, 60G50; Secondary: 47B15, 37A25, 37A30 quenched central limit theorem, Z d -action, random walk in random scenery, selfintersections of a r.w., toral automorphisms, S-unit, cumulant

We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random eld (r.f.) along a 2d-random walk in dierent situations: when the r.f. is iid with a second order moment (random sceneries), or when it is generated by the action of commuting automorphisms of a torus. We consider also a quenched version of the FCLT when the random walk is replaced by a Lorentz process in the random scenery.

Introduction

Let X = (X ) ∈Z d , d ≥ 1, be a strictly stationary real random eld (r.f.), where the X 's have zero mean and nite second moment. The r.f. can be represented in terms of dynamical system as X = T f , where (E, A, µ) is a probability space, T 1 , ..., T d are commuting measure preserving maps on (E, A, µ) and f is in L 2 (E, A, µ). 1 2 Let w = (w n ) n≥1 be a sequence of weights (or summation sequence), that is for each n a function ∈ Z d → w n ( ) ∈ R, with 0 < ∈Z d |w n ( )| < +∞. A natural question is the asymptotic normality in distribution of the self-normalized sums

∈Z d w n ( ) f (T x)/ ∈Z d
w n ( ) T f 2 and the estimation of the normalization factor. A stronger property, for some models, is the validity of a functional central limit theorem (FCLT).

Previously ( [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]), we have considered quenched central limit theorems for summation along a random walk, as well as summation on a sequence of sets in Z d . In a forthcoming paper, the FCLT for summation over sets will be presented.

The present paper is about the random walk case and specially the 2-dimensional random walk, the case of d-dimensional random walks being easier for d > 2. We show a FCLT in dierent models for the sums along a r.w. for almost all realizations of the r.w. (quenched FCLT).

One of these models is the random walk in random sceneries, i.e., the sums along a r.w. of a 2-d random eld of iid r.v.s with a moment of order 2. This improves a result of [START_REF] Guillotin-Plantard | A quenched functional central limit theorem for planar random walks in random sceneries[END_REF] which uses a slightly stronger moment condition. Our proof is short and self-contained. The same method can be used when the usual random walk is replaced by a plane Lorentz process (generated by a periodic billiard with dispersive obstacles) as in [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF]. A key step in the proof is then the law of large numbers shown in [START_REF] Pène | Self-intersections of trajectories of the Lorentz process[END_REF] for the self-intersection of the billiard map. The random sceneries can be also replaced by a random eld which is no more iid, but generated by an algebraically dened Z 2 -dynamical system. In this framework, we consider algebraic actions on tori by commuting automorphisms.

Tightness of the process is one of the main step of the proof of a FCLT. In the framework of sums along a random walk, our purpose is to present two dierent situations, independent and algebraic, as an illustration of two methods: one relying on the maximal inequality for associated r.v.s as shown by Newman and Wright [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF], the other on norm estimates for the maximum of partial sums as in 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] and others authors.

In Section 1 we gather results about the variance for the sums along a random walk. The independent case is presented in Section 2. Some general facts on the FCLT are recalled in Section 3, then applied to moving averages in Section 4 and to algebraic models in Section 5. 1 Underlined letters represent elements of Z d or T d . We write = ( 1 , ..., d ) and T f (x) = f (T 1 1. Summation along a r.w. and variance

Everywhere we assume (or prove) the absolute summability of the series of decorrelations:

k∈Z d | X T k f f dµ| < ∞, (1)
an hypothesis which implies existence and continuity of the spectral density associated to f , i.e., existence of a function ϕ f ∈ C(T d ) such that

X T k f f dµ = T d e 2πi k,t ϕ f dt, ∀k ∈ Z d . (2)
A method of summation is given by random walks (r.w.). If (Z n ) n≥0 is a random walk starting from 0 on Z d , the associated ergodic sums along the orbits of the random walk are

n-1 k=0 T Z k (ω) f = ∈Z d
w n (ω, ) T f, with w n (ω, ) = #{k < n : Z k (ω) = }.

(

) 3
Remark. Summation along the orbits of the random walk diers from summation over the range of the random walk. It has been shown ( [START_REF] Deligiannidis | Relative complexity of random walks in random scenery in the absence of a weak invariance principle for the local times[END_REF]) that the range of the random walk (Z n )

has the Følner property, so that summation over the range of the random walk yields a CLT.

Nevertheless, a functional CLT for summation over the range is a question.

1.1. Variance. First let us recall some results on the variance E | ∈Z d w n ( ) T f | 2 dµ which will be useful for the FCLT. Its computation is based on the normalized non-negative kernel

K(w n )(t) = | ∈Z d w n ( )e 2πi ,t | 2 ∈Z d |w n ( )| 2 , t ∈ T d . (4)
We say that w = (w n ) is ξ-regular, where ξ is a probability measure on T d , if (K(w n )) n≥1 weakly converges to ξ:

lim n→∞ T d K(w n ) ϕ dt = ξ(ϕ) for every continuous ϕ on T d , or equivalently if ξ(p) = lim n→∞ K(w n )(t) e -2πi p,t dt, ∀p ∈ Z d .
Under Condition (1), the asymptotic variance for f is then

σ 2 w (f ) := lim n ∈Z d w n ( ) T f 2 2 / ∈Z d |w n ( )| 2 = ξ(ϕ f ). (5)
In what follows, we will deal with examples which are δ 0 -regular. For examples of summation along a random walk which are ξ-regular with ξ = δ 0 , see for instance [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF].

Remark 1.1. If ξ is a probability measure on T d , f → (ξ(ϕ f )) 1 2
satises the triangular inequality. Indeed, for p(t) = a e 2πi ,t a trigonometric polynomial, we have:

|p(t)| 2 ϕ f (t) dt = a T f 2 2
, by denition of the spectral density; hence, by the triangular inequality,

( |p(t)| 2 ϕ f +g (t) dt) 1 2 ≤ ( |p(t)| 2 ϕ f (t) dt) 1 2 + ( |p(t)| 2 ϕ g (t) dt) 1 2 . It follows ϕ 1 2 f +g ≤ ϕ 1 2 f + ϕ 1 2 g , if ϕ f +g , ϕ f , ϕ g are continuous; hence: ϕ f +g ≤ ϕ f + ϕ g + 2ϕ 1 2 f ϕ 1 2 g , which implies ξ(ϕ f +g ) ≤ ξ(ϕ f ) + ξ(ϕ g ) + 2ξ(ϕ 1 2 f ϕ 1 2 g ) ≤ ξ(ϕ f ) + ξ(ϕ g ) + 2(ξ(ϕ f )) 1 2 (ξ(ϕ g ) 1 2 .

Question of non-degeneracy

Condition [START_REF] Billingsley | Convergence of probability measures, 2d edition[END_REF] implies that, for any δ 0 -regular summation sequence, the following conditions are equivalent: nullity of the asymptotic variance, ϕ f (0) = 0, k∈Z d T k f f = 0.

A function f is called a mixed coboundary if there exists measurable functions g i , i = 1, ..., d, such that f = d i=1 (I -T i )g i . In the example of a Z d -action by commuting algebraic automorphisms of a torus, for a class of regular function, the nullity of the asymptotic variance occurs if and only if f is a mixed coboundary. (See [START_REF] Cohen | Central limit theorem for commutative semigroups of toral endomorphisms[END_REF]).

1.2. Sums along random walks.

First we recall some denitions and notations.

Let (ζ i , i = 0, 1, ...) be a sequence of i.i.d. random vectors on a probability space (Ω, P) with values in Z d . The corresponding 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.
We suppose Z to be aperiodic 3 , with 0 mean, nite variance and (nonsingular) covariance matrix Σ. (For random walks, see [START_REF] Spitzer | Principles of random walk[END_REF].)

The space (Ω, P) can be viewed as (Z d ) Z equipped with a product measure and 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 holds.
Given a random eld (X , ∈ Z d ), we form the process n-1 k=0 X Z k (ω) obtained by summing along the r.w. (Z n ). If the random eld is represented as X = T f , it reads:

S ω n f = n-1 i=0 T Z i (ω) f = ∈Z d w n (ω, ) T f, with w n (ω, ) = 0≤i<n 1 Z i (ω)= . (6)
Summing along the random walk amounts to x ω in the ergodic sums of the skew product:

(ω, x) → T ζ 0 (ω, x) = (θω, T ζ 0 (ω) x) on Ω × E. Putting 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 i=0 F (T i ζ 0 (ω, x)) = n-1 i=0 f (T Z i (ω) x) = (S ω n f )(x). (7)
If we consider the r.v. S n F (ω, x) as dened on Ω × E endowed with the probability P × µ, a limit theorem is sometimes called an annealed limit theorem. We can also x ω ∈ Ω. A limit theorem in distribution (with respect to the measure µ on E) obtained for P-a.e. ω is called quenched.

We will consider the case where Z is a r.w. in Z 2 . In this case, (Z n ) is recurrent and a non standard normalization occurs in the CLT for sums along Z n .

1.2.1. On the number of self-intersections of a r.w.

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} (8)
3 i.e., we suppose that the subgroup generated in Z d by H := { :

P(ζ 0 = ) > 0} is Z d .
is non negative and increases when I or J increases for the inclusion order.

We write simply V (ω, I, p) for I = J, V (ω, I) for V (ω, I, 0) and V n (ω

) for V (ω, [0, n[). Hence V n (ω) = #{0 ≤ u, v < n : Z u (ω) = Z v (ω)} is the number of self-intersections starting from 0. Observe that V (ω, J) = ∈Z 2 w(ω, J, ) 2 . In particular V n (ω) = ∈Z 2 w 2 n (ω, ). Note also that V (ω, [b, b + k[) = V (θ b ω, [0, k[) = V k (θ b ω). 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(θ i+nA ω) - v-1 i=0 ζ(θ i+nA ω) = p} = V (θ nA ω, [0, n(B -A)], p).
For d = 2, there are C 0 , C nite positive constants 5 such that the following laws of large numbers hold (see: [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF] Lemma 2.6 for (9), [START_REF] Lewis | A law of the iterated logarithm for random walk in random scenery with deterministic normalizers[END_REF] step 1 in the proof of Proposition 1.4 for (10), and [4] Theorem 3.13 for (11)):

E(V n ) ∼ C 0 n ln n, Var(V n ) ≤ Cn 2 , (9)
ϕ n (ω) := V n (ω) C 0 n ln n → 1, for a.e. ω, (10)
ϕ n (ω, p) := V (ω, [1, n], p) C 0 n ln n → 1, ∀p ∈ Z d , for a.e. ω. (11)
We denote by Ω 0 the set of full probablity of ω's for which (10) and (11) hold. We have

V n (ω) ≤ K(ω) n ln 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 ln n, for ω ∈ Ω 0 .

(

) 13
Recall that (11) shows the δ 0 -regularity of the summation sequence along the random walk Z for a.e. ω (cf. [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]): if f has a continuous spectral density ϕ f ,

(C 0 n ln n) -1 n k=0 T Z k (ω) f 2 2 = (C 0 n ln n) -1 | n k=0 e 2πi Z k (ω),t | 2 ϕ f (t) dt → ϕ f (0). (14)
Before a preliminary lemma, let us introduce some more notations. For J ⊂ N, we put: 4 For simplicity, in the formulas below, we write nA, nB where we should write nA or nA + 1, nB . By convention θ t means θ t and the equalities below are true up to some additional quantities which are bounded independently from A, B, n.

U (m) (ω, J) := w(ω, J, ) m , U (m) n (ω) := w n (ω, ) m ; hence w n (ω, ) = w(ω, [0, n -1], ), V n (ω) = U (2) n (ω) = ∈Z d w 2 n ( ). For 1 , 2 , 3 ∈ Z d , we put W n (ω, 1 , 2 , 3 ) := (15) #{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 }.
5 If the r.w. is strongly aperiodic, C 0 = (π √ det Σ) -1 . For a general aperiodic r.w. in Z 2 , see for instance Theorem 5.1 in [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF].

By [2, Lemma 2.5] (see also [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]Proposition 2.9]) we have, for every ε > 0, sup ∈Z 2 w n (ω, ) = o(n ε ), for a.e. ω. [START_REF] Fréchet | A proof of the generalized second limit theorem in the theory of probability[END_REF] Therefore we have: for every ε > 0, there is C(ω) such that

U (m) n (ω) = w n (ω, ) m ≤ C(ω) n 1+ε . (17)
A more precise bound is given by the lemma:

Lemma 1.2. 1) For m ≥ 1, there is a positive integrable function K m such that

U (m) n (ω) = w n (ω, ) m ≤ K m (ω) n (ln n) m+1 , ∀n ≥ 1. (18)
2) There exists a positive integrable function C 1 such that

W n (ω, 1 , 2 , 3 ) ≤ C 1 (ω) n (ln n) 5 , ∀n ≥ 1. (19)
Proof. 1) For m = 1, 2, recall that w n (ω, ) = n and w n (ω, ) 2 ∼ C 0 n ln n. For m ≥ 3, let us show the bound [START_REF] Leonov | The use of the characteristic functional and semi-invariants in the ergodic theory of stationary processes[END_REF], which will be sucient for what is needed. We have

w n (ω, ) m = ( 0≤i<n 1 Z i (ω)= ) m = [ 0≤i 1 ,i 2 ,...,im<n 1 Z i 1 (ω)= 1 Z i 2 (ω)= . . . 1 Z im (ω)= ].
Since the terms in the above sum with equality between indices (hence for smaller values of m) can be treated by induction, it suces to bound

0≤i 1 <i 2 <•••<im<n 1 Z i 1 (ω)= 1 Z i 2 (ω)= . . . 1 Z im (ω)= = 0≤i 1 <i 2 <•••<im<n 1 Z i 1 (ω)= 1 Z i 2 (ω)-Z i 1 (ω)=0 1 Z i 3 (ω)-Z i 2 (ω)=0 • • • 1 Z im (ω)-Z i m-1 (ω)=0 .
Denote by Ψ n (ω) the above sum. Using independence and the local limit theorem for the random walk, we nd that the expectation of Ψ n is bounded by

C 0≤j 1 , j 2 , ••• , jm<n j -1 2 j -1 3 • • • j -1 m ≤ C n (ln n) m-1 . It follows: ∞ p=1 2 -p (ln 2 p ) -(1+m) Ψ 2 p dP < ∞; therefore, Ψ 2 p (ω) ≤ K(ω) 2 p (ln 2 p ) 1+m , with K := ∞ p=1 2 -p (ln 2 p ) -(1+m) Ψ 2 p ∈ L 1 (P).
Let p n be such that: 2 pn-1 ≤ n < 2 pn . Since Ψ n is increasing with n, we obtain:

Ψ n (ω) ≤ Ψ 2 pn (ω) ≤ K(ω) 2 pn (ln 2 pn ) 1+m ≤ K(ω) 2n(ln 2n) 1+m ≤ K (ω) n(ln n) 1+m .
2) The proof is similar to that of 1).

Study of the 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 1.3. 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]. Let n(δ) ≥ n 1 (δ) be such that k r+1 -k r ≤ δn(δ) for r ≤ n 1 (δ). Altogether, we get 0 < k r+1 -k r ≤ δn, ∀r ≤ n, if n ≥ n(δ).
The lemma implies: Lemma 1.4. 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 a.e. ω, 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, where c = P(Λ) -1 ; 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 1.3 implies 1). For 2), we select an increasing sequence of visit times to the set Λ satisfying the prescribed conditions by eliminating successive times at a distance < 1 2 δn.

Asymptotic orthogonality of the cross terms

We show the asymptotic orthogonality of the cross terms: for 0

< A < B < C < D < 1, ( 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. (20)
The above integral is the non negative self-intersection quantity:

V (ω, [nA, nB], [nC, nD], p). By (8), V (ω, I, J, p) increases when I or J increases. Hence, it suces to show (20) for the intervals [1, nA], [nA, n].
The proof below is based on (11) and ( 13).

Lemma 1.5. 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 ln n = 1, for A ∈]0, 1[, B = 1 -A; (21) 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. (22)
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 .
We will apply Lemma 1.4 to Λ(L(δ j ), δ j ) for each j, where (δ j ) is a sequence tending to 0, therefore getting a set ω's of full P-measure. The set Ω is the intersection of this set 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] Lewis | A law of the iterated logarithm for random walk in random scenery with deterministic normalizers[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. ( 23
)
Let us prove that for an absolute constant C 1 , for every δ, for n big enough:

ϕ nB (θ nA ω, p) = V (ω, [nA, n], p) C 0 nB ln n ∈ [1 -C 1 δ, 1 + C 1 δ]. (24)
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 ln n, C -1 0 V (ω, [1, nA], p) = (1 + ε n ) nA ln n. With B = 1 -A, this implies by (23) V (ω, [nA, n], p) C 0 nB ln n ≤ (1 + ε n ) n ln n -(1 + ε n ) nA ln n nB ln n ≤ 1 + |ε n | B + |ε n |A B ≤ 1 + 1 + A B δ.
Lower bound: We apply Lemma 1.4

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. Moreover, since ω ∈ Ω 0 , there is L such that ϕ n (ω, p) -1 ∈ [-δ, +δ] for n ≥ L.
We have, with

|δ n | ≤ δ, C -1 0 V (ω, [n A , n], p) ≥ (1 -δ n )(n -n A ) ln(n -n A ) = (1 -δ n )(nB -nAρ n ) ln(nB -nAρ n ). It follows, for δ (hence ρ n ) small: V (ω, [n A , n], p) C 0 (1 -δ n ) nB ln(nB) ≥ (nB -nAρ n ) ln(nB -nAρ n ) nB ln(nB) = (B -Aρ n ) [ln(nB) + ln(1 -A B ρ n )] B ln(nB) ≥ (1 - A B ρ n ) -2(1 - A B ρ n ) A B ρ n ln(nB) ≥ 1 - A B ρ n -2 A B ρ n ln(nB) ≥ 1 -B -1 δ(1 + 2 ln(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 ln(nB) ≥ 2, we have

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

This shows the lower bound and altogether with the upper bound we get [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF].

3) Proof of [START_REF] Móricz | Moment inequalities and the strong laws of large numbers[END_REF]. Let δ > 0. According to (23) and ( 24), for n big enough, we have

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 ln n -(1 + ε n ) nA ln n -(1 + ε n ) n(1 -A) ln n ≤ 2C 0 δ n ln n.
In the next section, we will need to compute the asymptotic variance for s j=0 a j nt j k=nt j-1 T Z k (ω) f , where a 1 , ..., a s are real numbers and 0 = t

0 < t 1 < ... < t s-1 < t s = 1 is a subdivision of [0, 1].
Its value is given by the following corollary of (20) and Lemma 1.5: Lemma 1.6. Assume that f has a continuous spectral density ϕ f . For a.e. ω we have

(C 0 n ln 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 ). (25)
Proof. 1) Recall that proving [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF] amount to prove

(C 0 n ln 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 (20) for a nite set of characters e -2πi p,u .

Using [START_REF] Esary | Association of Random Variables, with Applications[END_REF] for the asymptotic variance starting from 0, we have

(C 0 n ln n) -1 tn k=0 T Z k (ω) f 2 2 → tρ(0), for t ∈]0, 1[. By Lemma 1.5, (C 0 n ln n) -1 tn k= sn T Z k (ω) f 2 2 → (t -s) ρ(0), for 0 < s < t < 1,
and, expanding the square and using that the cross terms are asymptotically negligible,

(C 0 n ln n) -1 | s j=1 a j nt j k=nt j-1 e 2πi Z k (ω),u | 2 ρ(u) du behaves like (C 0 n ln 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 [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF] 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 ln 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 ln 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 ln 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 ln n) -1 V n (ω) + s j=1 a 2 j (t j -t j-1
), which is bounded uniformly with respect to n. Therefore we can conclude For a general continuous spectral density by step 1).

Remarks 1.7. 1) In Lemma 1.4, 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), the con- clusion (25) of the lemma is a consequence of the law of large numbers for the number of self-intersections Vn(ω)

C 0 n ln n → 1.
The law of large numbers for V n (ω, p), p = 0 is not needed. 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 theorem [START_REF] Cramér | Some Theorems on Distribution Functions[END_REF] 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. (26)
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 (ω, x, t ) -Y n (ω, x, t)| ≥ ε) = 0. (27)
Now, let (Z n ) be a 2-dimensional centered random walk with a nite moment of order 2 as in Subsection 1.2. Recall the notation w n (ω, ) := n-1 k=0 1 Z k (ω)= . We use also the notation w(ω, J,

) := k∈J 1 Z k (ω)= , for J = [b, b + k[.
For a real stationary process 6 X = (X ) ∈Z 2 on (E, µ), we denote the sums along the random walk by

S ω n (x) := n i=1 X Z i (ω) (x) = n i=1 f (T Z i (ω) x) = w n (ω, ) f (T x), n ≥ 1,
A quenched FCLT is satised by the r.f. X = (X ) ∈Z 2 if for a.e. ω the functional central limit theorem holds for the process

(Y n (ω, x, t)) t∈[0,1] := S ω [nt] (x) √ n log n t∈[0,1] . ( 28
)
6 Recall that the process is denoted either by (X ) or by (T f ).

When (X ) is an iid random eld, the model is the so-called random walk in random scenery (RWRS). In the next section we consider rst this independent case, before other non independent models in the last sections.

2. Independent random eld 2.1. Random walk in random scenery.

Let (X (x)) ∈Z 2 = (T f (x)) ∈Z 2 be a random eld of centered i.i.d. real variables with E(X 2 0 ) = 1 and mean 0 on a space (E, µ).

We consider the random walk in random scenery S ω n (x) and the process dened by [START_REF] Spitzer | Principles of random walk[END_REF].

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, that is: with respect to the probability P × µ, the law of Y n converges weakly to the Wiener measure. A quenched FCLT under the assumption

E[|X 0 | 2 (log + |X 0 |) χ ] < ∞,
for some χ > 0, has been proved for (Y n (ω, x, t)) in [START_REF] Guillotin-Plantard | A quenched functional central limit theorem for planar random walks in random sceneries[END_REF], based on [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF], a result of E. Bolthausen and A-S. Sznitman (2002) and a truncation argument.

In this section, we will give a direct proof of the quenched FCLT for an iid r.f. as well as for a moving averages of an iid r.f. in Section 4, assuming only the existence of a moment of order 2. As in [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF] for the annealed FCLT, our proof follows the method of Newman and Wright [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF] for associated random variables.

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 all non-decreasing (in each coordinate) functions f, g : R n → R, we have, if the covariance exists:

Cov(f (X 1 , . . . , X n ), g(X 1 , . . . , X n )) ≥ 0.

An innite collection X 1 , X 2 , . . . of variables is said to be associated if every nite sub-collection is associated.

It is known that every subset of an associated family is associated. Moreover, every collection of non-decreasing functions of a family of associated random variables are associated. It follows that if (X k ) is an associated family, in particular independent, then (X Z k (ω) ) is an associated family 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 nt (x) √ n log n t∈[0,1]
satises a FCLT with asymptotic variance σ 2 = (π

√ det Σ) -1 .
Proof. 1) First, let us recall Lindberg's CLT. Let (W n,k ) rn k=1 be a triangular array of independent random variables with zero mean and nite variance. Let In our situation we consider sums of the form n-1

σ 2 n = rn k=1 W n,k 2
k=0 X Z k (ω) = ∈Z 2 w n (ω, )X , where w n (ω, ) = #{0 ≤ k < n : Z k (ω) = }.
In order to apply the Lindeberg CLT we put the following (for a xed ω): X n, = w n (ω, )X . For each n, #{ : w n (ω, ) = 0} ≤ n, which ensures that r n is nite for every n.

Here, Lindeberg's condition reads (with σ 2 n = w 2 n ( ) = V n ) (recall that we asume E(X 2 0 ) = 1):

∈Z 2 1 V n w 2 n ( )X 2 1 {wn( )X ≥ σn} dµ = 1 V n X 2 0 ∈Z 2 w 2 n ( ) 1 {w 2 n ( )X 2 0 ≥ 2 Vn} dµ,
and by the a.e. uniform estimate (over ) given by ( 16) and lim n Vn C 0 n log n = 1 for a.e. ω, w 2 n ( )/V n → n 0, uniformly over , the condition is satised. Eventually, we get that for a.e. ω, the sequence 1

√ Vn n-1 k=0 X Z k (ω)
converges in law toward a normal law, but we know that lim n V n /(C 0 n log n) = 1 for a.e. ω. So, replacing the normalization by √ C 0 n log n and obtain a CLT to a standard law.

For the nite dimensional distributions, we apply the Cramér-Wold device. We know that we have for a.e. ω asymptotic orthogonality for sums over disjoint intervals. Moreover, for a.e. ω and for every subdivision of [0, 1], we have

(C 0 n log n) -1 s j=1 a j nt j k= nt j-1 X Z k (ω) 2 2 → n s j=1 a 2 j (t j -t j-1 ),
In this case, because of the asymptotic orthogonality, it is enough (since the r.w. are associated (cf. [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF]) to prove the convergence of

1 √ C 0 n log n t j n k= t j-1 n X Z k (ω) towards N (0, √ t j -t j-1
), for every j. We put X n = w( , [nt j-1 , nt j ])X , where w( , [nt j-1 , nt j ]) = #{nt j-1 ≤ k < nt j : Z k = }. We note that for a.e. ω 1 C 0 n log n w 2 (ω, , [nt j-1 , nt j ]) → n t j -t j-1 , for every j. We see that Lindeberg's condition holds for this setting as before.

2) Tightness. The following is proved in the proof of Theorem 3 in [START_REF] Newman | An invariance principle for certain dependent sequences[END_REF]:

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

µ( max 1≤k≤n |S k | ≥ λ E(S 2 n )) ≤ 2µ(|S n | ≥ (λ - √ 2) E(S 2 n )). (29)
Inequality (29) can be applied to U k = X Z k (ω) for every xed ω, as well as to the sums S J = k∈J X Z k starting from any point in [0, n]. We also note that E(S 2 J ) = X 0 2 2 V (ω, J). First, let us assume that E(X 4 0 ) < ∞. We have then:

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 ≤ 3E(X 2 0 ) 2 ( w(ω, J, ) 2 ) 2 + E(X 4 0 ) w(ω, J, ) 4 = 3E(X 2 0 ) 2 V (ω, J) 2 + E(X 4 0 ) U (4) (ω, J). (30)
By [START_REF] Guillotin-Plantard | A quenched functional central limit theorem for planar random walks in random sceneries[END_REF] there is a positive a.e. nite function K 0 such that

U (4) n (ω) = w 4 n (ω, ) ≤ K 0 (ω) n 2 .
(31)

Let K(ω) := max(K(ω), K 0 (ω)). Let C be a constant > 0 such that µ{ω : K(ω) ≤ C} > 0.

Using Lemma 1.4, for a given n and δ ∈]0, 1[, we can choose good times 0 = ρ 1 < ρ 2 < ... < ρ v ≤ n < ρ v+1 , with v < 2/δ, such that θ ρ i ω ∈ {ω : K(ω) ≤ C} 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 and J i = [ρ i-1 , ρ i-1 + 1, . . . , ρ i ), m i = ρ i+1 -ρ i ≤ 3 2 δn. We have ρ i j=ρ i-1 X Z j (ω) 4 4 = V 2 (ω, J i ) + X 0 4 4 U (4) (ω, J i ) ≤ C n 2 δ 2 log 2 (nδ) + C X 0 4 4 n 2 δ 2 , ∀i. There is K 1 > 1 such that m i log m i V (ω,J i ) ∈ [K -1 1 , K 1 ].
Now, using (29) and a classical method (cf. [START_REF] Billingsley | Convergence of probability measures, 2d edition[END_REF]), we get:

µ( sup |t -t|≤δ |Y n (t) -Y n (s)| ≥ 3ε) ≤ v i=1 µ( sup t i-1 ≤s≤t i | sn k=[t i-1 n X Z k (ω) | ≥ ε n log n) ≤ v i=1 µ( sup ρ i-1 ≤k≤ρ i | k j=ρ i-1 X Z j (ω) | ≥ ε 2 3 m i δ log 2 3 m i δ ) ≤ v i=1 µ( sup ρ i-1 ≤k≤ρ i | k j=ρ i-1 X Z j (ω) | ≥ λ 2 3 m i log 2 3 m i V (ω, J i ) ) V (ω, J i ) ≤ 2 v i=1 µ(| ρ i j=ρ i-1 X Z j (ω) | ≥ (λ 2 3 m i log 2 3 m i V (ω, J i ) - √ 2) V (ω, J i )).
We estimate the probabilities inside the above sum. By Chebyshev's inequality (for moment of order 4), we have, for λ (hence δ small enough) such that λK

-1/2 1 - √ 2 > 1 2 λK -1/2 1 : µ | ρ i j=ρ i-1 X Z j (ω) | ≥ (λ 2 3 m i log 2 3 m i V (ω, J i ) - √ 2) V (ω, J i ) ≤ C X 0 4 4 m 2 i log 2 m i V 2 (ω, J i )(λ 2 3 m i log 2 3 m i V (ω,J i ) - √ 2) 4 ≤ 16CK 4 1 X 0 4 4 λ 4 .
We get with C 1 = 16CK

|Y n (t) -Y n (s)| ≥ 3ε) ≤ C 1 X 0 4 4 2 δ δ 2 ε 4 = 2C 1 X 0 4 4 δ ε 4 .
Now we use a truncation. For a given 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 (29) holds for both sums, except that for the unbounded part of the truncation we do not have moment of order 4. We use Chebyshev's inequality (for moment of order 2) to control the unbounded truncated part:

µ | m i j=m i-1 XL Z j (ω) | ≥ (λ 2 3 m i log 2 3 m i V (J i , ω) - √ 2) V (J i , ω) ≤ C X 0 -XL 0 2 2 m i log m i V (J i , ω)(λ 2 3 m i log 2 3 m i V (J i ,ω) - √ 2) 2
.

Hence, for n and λ big enough, the sum is comparable for some constant C with

C v i=1 XL 0 2 2 λ 2 ≤ C 1 δ δ XL 0 2 2 ε 2 = C ε -2 XL 0 2 2 .
Using the inequality µ(|f

+ g| ≥ ε) ≤ µ(|f | ≥ ε 2 ) + µ(|g| ≥ ε 2 )
, we obtain the bound:

µ( sup |t -t|≤δ |Y n (t ) -Y n (t)| ≥ 3ε) ≤ 16C 1 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 16C 1 L 4 δ ε 4 ≤ 1 2 η.

A model based on the Lorentz process.

In this subsection we sketch briey how to obtain an version of a FCLT where 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 detail on this model.

Let be given a billiard table in the plane, union of Z 2 -periodically distributed sets, the 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 suppose (dispersing billiard) 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 assume a 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 occur. An annealed FCLT for W n has been shown by F. Pène ([25]): there exists β 0 > 0 such that W [nt] β 0 n lg n converges weakly to the standard Wiener process.

To extend the result to a quenched version, we use [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF]Proposition 7], in place of Inequality [START_REF] Guillotin-Plantard | A quenched functional central limit theorem for planar random walks in random sceneries[END_REF] for the r.w., and [START_REF] Pène | Self-intersections of trajectories of the Lorentz process[END_REF]Corollary 4] (the main and most dicult step), which gives a law of large numbers for the self-intersections of the billiard transformation replacing (10). Then, Remarks 1.7 and the preceding method for the r.w. in random sceneries yield the quenched version of the FCLT for this model.

Other methods for the FCLT

For the models of r.f. in Sections 4 and 5, we need to introduce some tools, in particular for the tightness, for which a method based on 4th-moment is used. This is the content of this section.

Cumulants and CLT.

The method of cumulants recalled below can be helpful to prove the CLT in dynamical systems.

In 1960, Leonov ([18], [START_REF] Leonov | On the central limit theorem for ergodic endomorphisms of compact commutative groups (Russian)[END_REF]) applied it to a single algebraic endomorphism of a compact abelian group. In [START_REF] Cohen | Almost mixing of all orders and CLT for some Z d -actions on subgroups of F Z d p[END_REF], [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF], it was applied to multidimensional actions by algebraic endomorphisms.

Moments and cumulants

In this subsection, the random variables are assumed to be uniformly bounded and centered.

Let (X 1 , ..., X r ) be a random vector. For I = {i 1 , ..., i p } ⊂ J r := {1, ..., r}, let m(I) = m(i 1 , ..., i p ) := E(X i 1 • • • X ip ). A denition of the cumulant of (X 1 , ..., X r ) using the moments is

C(X 1 , ..., X r ) = Π={I 1 ,I 2 ,...,I p(Π) }∈P (-1) p(Π)-1 (p(Π) -1)! m(I 1 ) • • • m(I p(Π) ), (32)
where Π = {I 1 , I 2 , ..., I p(Π) } runs through the set P of partitions of J r = {1, ..., r} into nonempty subsets and p(Π) is the number of elements of Π.

For example, if r = 4, the cumulant of centered r.v.s is C(X 1 , X 2 , X 3 , X 4 ) = E(X 1 X 2 X 3 X 4 )-[E(X 1 X 2 ) E(X 3 X 4 )+E(X 1 X 3 ) E(X 2 X 4 )+E(X 1 X 4 ) E(X 2 X 3 )]. Putting s(I) := C(X i 1 , ..., X ip ) for I = {i 1 , ..., i p }, we have E(X 1 • • • X r ) = Π={I 1 ,I 2 ,...,I p(Π) }∈P s(I 1 ) • • • s(I p(Π) ). ( 33
)
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

C (2) (Y ) = Y 2 2 , C (4) (Y ) = E(Y 4 ) -3E(Y 2 ) 2 , E(Y 4 ) = 3E(Y 2 ) 2 + C (4) (Y ). If (X ) ∈Z d = (T f ) is a stationary random eld, we put C f ( 1 , ..., r ) = C(X 1 , ..., X r ).
The following theorem (cf. [START_REF] Leonov | On the central limit theorem for ergodic endomorphisms of compact commutative groups (Russian)[END_REF]Th. 7], [4, Th. 6.2]) gives a criterium in terms of cumulants to prove a CLT and the convergence of the normalised moments toward those of the normal law.

is asymptotically distributed as N (0, J s ), where J s is the s-dimensional diagonal matrix with diagonal ζ j (ϕ f ).

Proof. The hypothesis (37) implies

( j a 2 j ∈Z d w j n ( ) 2 ) -1 | ∈Z d j a j w j n ( ) e 2πi ,t | 2 weakly -→ n→∞ ( j a 2 j ) -1 j a 2 j ξ j . (39)
For s non zero real parameters a 1 , ..., a s , let (w a 1 ,...,as n ) n≥1 be dened by w a 1 ,...,as n ( ) = a 1 w 1 n ( ) + ... + a s w s n ( ).

By the Cramér-Wold theorem, for the conclusion of the theorem, it suces to show that the process ∈Z d w a 1 ,...,as n ( ) T f after normalization satises the CLT:

∈Z d w a 1 ,...,as n ( )T f (a 2 1 ∈Z d w 1 n ( ) 2 + ... + a 2 s ∈Z d w s n ( ) 2 ) 1 2 ⇒ N (0, s j=1 a 2 j ξ j (ϕ f )/ s j=1 a 2 i ). (40)
By (38), the sum

i 1 ,...,ir∈{1,...,s} r ( 1 ,..., r ) ∈(Z d ) r w i 1 n ( 1 )...w ir n ( r ) C(X 1 , ..., X r ) satises (34) and
the result follows from Theorem 3.1.

The following lemma will be useful in the proof of the asymptotic normality for the nite dimensional distributions.

Let (w n ) n≥1 be a summation sequence on Z d which is ξ-regular.

For f ∈ L 2 (µ), we put σ n (f ) := w n ( ) T f 2 . Lemma 3.3. Let f, f k , k = 1, 2, ..., ∈ L 2 (µ) satisfying (1) such that ϕ f -f k ∞ → 0. Then σ n (f k ) -1 ∈Z d w n ( ) T f k distrib =⇒ n→∞ N (0, 1), ∀k ≥ 1, implies σ n (f ) -1 ∈Z d w n ( ) T f distrib =⇒ n→∞ N (0, 1).
Proof. Let (ε k ) be a sequence of positive numbers tending to 0, such that ϕ f -f k ∞ ≤ ε k . 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 the ξ-regularity of (w n ), we have:

( ∈Z d w 2 n ( )) -1 ∈Z d w n ( ) T f 2 2 = T d wn ϕ f dt → n→∞ ξ(ϕ f ).
We can suppose ξ(ϕ f ) > 0, since otherwise the limiting distribution is δ 0 . We have ξ(ϕ f -f k ) → 0. It follows that ξ(ϕ f k ) = 0 for k big enough (cf. Remark 1.1).

We have:

U k n distr =⇒ 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 wn ϕ f -f k dt = ξ(ϕ f -f k ) ≤ ε 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 the conclusion U n distr =⇒ n→∞ N (0, ξ(ϕ f )) follows from [1, Theorem 3.2].

A sucient condition for tightness.

A nonnegative function

G 0 = (G 0 (b, n), b, n ≥ 0) is said to be 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. (41)
Let (W k ) be a sequence of real or complex r.v. on a probability space (E, µ). With the notation

S b,n = b+n r=b+1 W r , M b,n = max 1≤k≤n |S b,k |,
we re-state the following result of Móricz as it is used here.

Theorem 3.4. (Móricz, [START_REF] Móricz | Moment inequalities and the strong laws of large numbers[END_REF]) Suppose that there exists G 0 satisfying (41) such that

E µ (|S b,n | 4 ) ≤ G 0 (b, n) 2 , ∀b ≥ 0, ∀n ≥ 1. (42)
Then, with the constant

C max = (1 -2 -1 4 ) -4 , E µ (|M b,n | 4 ) ≤ C max G 0 (b, n) 2 , ∀b ≥ 0, ∀n ≥ 1. (43)
The maximal inequality (43) gives a criterium of tightness: Proposition 3.5. 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 ln k, H(ω, b, k) ≤ K 2 (θ b ω) k (ln k) γ , G(ω, b, k) ≥ k. ( 44
)
Then there is a constant C 1 and for every δ ∈]0, 1] an integer N (δ) such that, for every ε > 0, for every stationary r.f. X = (X ) ∈Z d under the conditions

lim n→∞ µ( A n √ n ln n ≥ ε) = 0, where A n := n ∆n max u=0 ∆n max k=1 | u∆n+k-1 j=u∆n X Z j (ω) |, with ∆ n = n 1 2 (log n) 2 . ( 45
) E µ (|S ω,X J | 4 ) ≤ G(ω, b, k) 2 + n 1 2 (ln n) -(γ+1) H(ω, b, k), ∀J = [b, b + k] ⊂ [1, n], for a.e. ω, (46)
the normalised sum

Y n (ω, x, t) = 1 √ n ln n [nt] j=1 X Z j (ω) (47)
satises for n ≥ N (δ):

µ(x ∈ E : sup |t -t|≤δ |Y n (ω, x, t ) -Y n (ω, x, t)| ≥ ε) ≤ C 1 ε -4 δ, for a.e. ω. (48)
Proof.

1) Let c ≥ 0, ν = ν n ≥ 1, ∆ n = n 1 2 (ln n) -2 ∈ [1, ν], L n = [ νn ∆n ], ν = ν n = [ νn ∆n ]∆ n +∆ n -1.
The integer ν n will be chosen later of order δn.

Since ν n ≤ ν n , 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-1 | 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-1 | u-1 r=0 (r+1)∆n-1 j=r∆n X Z j+c (ω) | + max 0≤u≤Ln, 1≤k≤∆n-1 | u∆n+k-1 j=u∆n X Z j+c (ω) | = max 0<u≤Ln | u∆n-1 j=0 X Z j+c (ω) | + max 0≤u≤Ln, 1≤k≤∆n-1 | u∆n+k-1 j=u∆n X Z j+c (ω) | = Ân + Ãn .
With Ân and Ãn respectively the rst and the second term above, this implies

µ( max 0≤k≤ν | k j=0 X Z j+c (ω) | ≥ ε √ n ln n) ≤ µ( Ân ≥ 1 2 ε √ n ln n) + µ( Ãn ≥ 1 2 ε √ n ln n). (49)
For Ãn , we get lim n µ( Ãn

≥ 1 2 ε √ n ln n) = 0 by (45). Therefore there is N 1 (δ) such that µ( Ãn ≥ 1 2 ε √ n ln n) ≤ δ, for n ≥ N 1 (δ).
For Ân we will apply Theorem 3.4 to W r = (r+1)∆n-1 j=r∆n

X Z c+j (ω) , with G 0 (b, k) := G(ω, c + b∆ n , k∆ n ) + (ln n) -(γ-1) H(ω, c + b∆ n , k∆ n ), (50)
Since G(ω, c

+ b∆ n , k∆ n ) ≥ G(ω, c + b∆ n , ∆ n ) ≥ ∆ n = n 1 2 (ln n) -2 , we have G 2 (ω, c + b∆ n , k∆ n ) + n 1 2 (ln n) -(γ+1) H(ω, c + b∆ n , k∆ n ) ≤ (G 0 (b, k)) 2 , for every interval [b, b + k∆ n [. Therefore E µ (| b+k r=b+1 W r | 4 ) = E µ (| (b+k+1)∆n-1 j=(b+1)∆n X Z c+j (ω) | 4 ) ≤ G 0 (b, k) 2 , ∀b ≥ 0, ∀n ≥ 1,
which implies by (43) of Theorem 3.4:

E µ ( max 1≤k≤p | (b+k+1)∆n-1 j=(b+1)∆n X Z c+j (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 (44), we get the bound

Ln max u=1 | u∆n j=0 X Z c+j (ω) | 4 4 ≤ C max [G(ω, c, L n ∆ n ) + (ln n) -γ+1 H(ω, c, L n ∆ n )] 2 ≤ C max K(θ c ω) 2 [L n ∆ n ln(L n ∆ n ) + (ln n) -(γ-1) (L n ∆ n )(ln(L n ∆ n )) γ ] 2 .
(51)

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 1.4 to Ω M . There is N 2 (δ) such that for n ≥ N 2 (δ), we can nd 0 = ρ

(n) 1 < ρ (n) 2 < ... < ρ (n) v ≤ n < ρ (n)
v+1 , a sequence of visit times of θ k ω in Ω M under the iteration of the shift θ,

such that 1 2 δn ≤ ρ (n) i+1 -ρ (n) i ≤ 3 2 δn and v < 2/δ. By construction, K(θ ρ (n) i ω) ≤ M, ∀i. With c = ρ (n) i , ν n = ν i,n = ρ (n) i+1 -ρ (n) i ≤ 3 2 δn, L (n) i = [ ν i,n
∆n ] (so that L

(n) i ∆ n ≤ δn), we deduce from the upper bound (51) (for n big enough):

L (n) i max u=1 | u∆n j=0 X Z ρ (n) i +j (ω) | 4 4 ≤ C max M [ν (n) i ln ν (n) i + (ln n) -γ+1 ν i,n (ln(ν i,n ) γ ] 2 ≤ C max M [ 3 2 δn ln(δn) + (ln n) -γ+1 δn(ln(δn)) γ ] 2 ≤ C max M [3δn ln n] 2 .
where in the last inequality we use 0 ≤ ln(δn) ≤ ln n, which holds if n ≥ δ -1 .

This implies for Ân (cf. ( 49)), for i = 1, ..., v, for a constant C:

µ( max 0≤u≤[ ν i ∆n ] | u∆n-1 j=0 X Z j+ρ i (ω) | ≥ 1 2 ε √ n ln n) ≤ 2C max M (3δn ln n) 2 ( 1 2 ε √ n ln n) 4 ≤ C ε -4 δ 2 .
Putting t i = ρ i /n, we obtain (48), i.e., for a given ε > 0 and n ≥ N (δ) for N (δ) big enough,

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 3.6. Let be given for s in a set of indices S a process X s = (X s ) ∈Z 2 satisfying (45) and (46) for each s, with the same G, H, γ. Then, if X = s a s X s with s |a s | ≤ 1, the r.f. X = (X ) satises these conditions and therefore the conclusion (48).

This follows immediately from Minkowski inequality. For example, we have for (46):

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 (ln n) -(γ+1) H(ω, b, k)] 1 4 ) 4 = ( s |a s |) 4 [G(ω, b, k) 2 + n 1 2 (ln n) -(γ+1) H(ω, b, k)].
4. Linear combination (moving averages) of iid bounded variables Let (X ) ∈Z 2 be a r.f. of centered i.i.d. real variables 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 ) ∈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 (Ξ ) ∈Z 2 is ϕ Ξ (t) = | a q e 2πi q,t | 2 . We assume that the asymptotic variance is > 0, a condition equivalent to q∈Z 2 a q = 0.

Using the method of associated r.v.s we obtain a quenched FCLT. An annealed FCLT can be shown with a proof along the same lines.

Theorem 4.1. 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 assume the random variables are bounded. We prove the CLT for 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 , follows by an approximation argument.

For F , there is M such that 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 1 F which is independent from the others in the collection T 1 F, ..., T r F (by niteness of S).

We put w u n (ω, ) := nt u+1 j=ntu 1 Z j = . Observe that w u n (ω, ) ≤ w n (ω, ).

Since sup

1 ,..., r |C(T 1 F, T 2 F, ..., T r F )| < ∞, we have | 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 .

For every ε > 0, there is C ε (ω) a.e. nite and such that sup w n (ω, ) ≤ C ε (ω) n ε (cf. ( 16)).

Hence:

∈Z d r k=1 w n (ω, , + j k ) ≤ C ε (ω) r-1 n ε(r-1) n = o(n r/2
), ∀r ≥ 3, and (34) is satised.

Using Lemma 3.3, the result can be extended to any sum s∈S a s X s , with s∈S |a s | < ∞. b) Now if we assume only the condition X 0 2 < ∞, we use a truncation argument and apply again Lemma 3.3.

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), for ∈ Z 2 , are associated, as well as q∈Z 2 a - q X -q (x), for ∈ Z 2 .

Therefore tightness can be proved separately for both processes. The proof is like the proof of the tightness in Theorem 2.2 and this implies tightness for the moving averages.

Algebraic endomorphisms

We consider a second type of example, generated by the action of two commuting automorphisms on tori. For the tightness, in this example we use the moment inequality on maxima and Proposition 3.5. This method could be used also in the independent model, but with a strengthening of the moment hypothesis.

N d -actions by endomorphisms on a compact abelian group

Let G be a compact abelian group with Haar measure µ. We assume that the action on G is mixing of all orders (this holds if it is totally ergodic and G is connected, which is the case of a totally ergodic action on a torus).

Let AC 0 (G) denote the class 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)| < +∞. Since the characters T χ for ∈ Z d are pairwise distinct, we have the inequalities

∈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 ϕ f of f is continuous and

ϕ f ∞ ≤ f 2 c . (52)
Proposition 5.1. If f is in AC 0 (G), the spectral density ϕ f is continuous on T ρ and for every ε > 0 there is a trigonometric polynomial P such that ϕ f -P ∞ ≤ ε. Proof. By (52), 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.

5.1. Matrices and automorphisms of the torus. Now we will restrict to the special case of matrices and endomorphisms of the torus G = T ρ .

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 H = T ρ identied with Z ρ (this is the action by the transposed of B, but since we compose commuting matrices, for simplicity we do not write the transposition). 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 A ∈ M * (ρ, Z) acts ergodically on (T ρ , µ) if and only if A has no eigenvalue root of unity.

Let (A 1 , ..., A d ) be a nite family of d commuting non singular matrices in Gl(ρ, Z) and A = A We obtain a Z d -action (A , ∈ Z d ) on (T ρ , µ), which is totally ergodic if and only if the dual action is free on Z ρ \{0}, or equivalently if A has no eigenvalue root of unity for = 0.

The composition with a function f dened on T ρ will be denoted f • A , or A f or T f . Explicit examples can be computed like the example below (cf. the book of H. Cohen on computational algebraic number theory [START_REF] Cohen | A course in computational algebraic number theory[END_REF]): Then there are nite constants B 1 , B 2 such that | A f, f | ≤ B 1 f 2 -α , ∀ = 0, the spectral density ϕ f is continuous,

∈Z d | A f, f | < ∞ and ϕ f ∞ ≤ B 2 f 2 .
For compact abelian groups which are connected ( [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 ( [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 last result at least in the case of automorphisms of a torus to a functional CLT.

Number of solutions

We use the following fundamental result on S-unit equations ( [START_REF] Schlickewei | S-unit equations over number elds[END_REF]):

Theorem 5.3. ([15, 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 ρ. For any (a 1 , ..., a r ) ∈ (K * ) r , the number A(a 1 , ..., a r , Γ) 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, satises the estimate A(a 1 , ..., a r , Γ) ≤ A(r, Γ r ) = exp((6r) 3r (ρ + 1)).

There is a decomposition of E = C ρ into vectorial subspaces C ρ = ⊕ k E k which are simultaneously invariant by Ãi , i = 1, ..., d, 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 easily from the fact that the commuting matrices A i have a common non trivial space W of eigenvectors, and then from an induction on the dimension of the vector space, applying the induction hypothesis to the action of the maps Ãi on the quotient E/W .

1

 1 {W n,k ≥εσn} dµ = 0, ∀ε > 0 (Lindeberg's condition), then 1 σn rn k=1 W n,k converges in law to the standard normal law.

1 1 .

 1 ..A 1 1 , for = ( 1 , ..., d ) ∈ Z d .

  Spectral density and rate of decorrelation for automorphisms of the torusA sucient condition for f to be in AC 0 (T ρ ) is the following decay of its Fourier coecients:| f (k)| = O( k -β ), with β > ρ. (53)The condition of regularity∃α > d : ∀δ > 0, f (. + τ 1 , • • • , . + τ ρ ) -f 2 ≤ C(f ) (ln 1 δ ) -α , ∀ |τ 1 |, ..., |τ ρ | ≤ δ (54) is also sucient if α > d (cf. ([[START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]). Recall also the following consequence for the continuity of the spectral density.Proposition 5.2. ([START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]) Let f be in L 2 0 (T ρ ) satisfying (54) for some α > d.

  1.3. Formulation of the quenched FCLT.

Let (Y n (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 with left limits, endowed with the uniform norm.

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 (denoting by =⇒ the convergence in distribution):

  The group of characters of G is denoted by Ĝ or H and the set of non trivial characters by Ĝ * or H * . The Fourier coecients of a function f in L 1 (G, µ) are c f (χ) := G χ f dµ, χ ∈ Ĝ. surjective endomorphism B of G denes a measure preserving transformation on (G, µ) and a dual injective endomorphism on Ĝ. For simplicity, we use the same notation for the actions on G and on Ĝ. (T 1 , ..., T d ) be a nite family of d commuting surjective endomorphisms of G and T = T1 1 ...T 1 1 , for = ( 1 , ..., d ) ∈ Z d .We obtain a Z d -action (T , ∈ Z d ) on G, which is totally ergodic if and only if the dual action is free. The composition with a function f dened on G is denoted T f .

	Every

Let

Proof. If ξ(ϕ f ) = 0 the asymptotic variance (for the normalisation as in ( 35)) is 0. Otherwise, we have lim n w n ( )

The multi-linearity of the cumulants implies:

Let Z (n) = w n ( ) X -1 2 w n ( ) X . It follows from ( 36) and (34):

By the formula linking moments and cumulants, the convergence of the moments follows from the convergence to 0 of the cumulants of order > 2. We apply to (Z (n) ) n≥1 a theorem of M.

Fréchet and J. Shohat [START_REF] Fréchet | A proof of the generalized second limit theorem in the theory of probability[END_REF] (expressed in terms of moments) and obtain (35).

Array of sequences and nite dimensional distributions

Let (w j n ) n≥1 , j = 1, ..., s, be ξ j -regular summation sequences, where the ξ j 's are probability measures on T d . Using Theorem 3.1, we are going to deduce from two conditions the asymptotic normality (after normalization) of the vectorial process

-asymptotic orthogonality 7 :

-convergence to 0 of the normalized cumulants of order ≥ 3:

Proposition 3.2. Under Conditions (37) and (38), the vectorial process

7 to shorten the notation, everywhere we write w j n ( ) 2 instead of (w j n ( )) 2

Let us now consider on the torus T ρ a character χ γ , x → exp(2πi γ, x ), where 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

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

Using the total ergodicity of the action of the endomorphisms associated to

We will apply Thoerem 5.3 to the multiplicative group generated by α j,k 0 , j = 1, ..., d.

Random walks and quenched CLT.

Our aim is to replace the model of i.i.d. variables (X( ), ∈ Z 2 ) discussed in Section 2 by

generated by an observable f on a torus T ρ under the action of commuting automorphisms.

More precisely, we consider

The following result (cf. [4, Lemma 6.6] gives a sucient condition for the asymptotic nullity of the cumulants.

Proposition 5.4. If (T , ∈ N d ) is mixing of order r ≥ 2, then, for any f ∈ L ∞ 0 (X), lim

The lemma does not give the quantitative estimate which is necessary to use the sucient condition (35). Nevertheless, it suces for a N d -action by endomorphisms of a compact abelian group if this action is mixing of order r ≥ 2 and f is a trigonometric polynomial.

The following quenched FCLT extends the CLT proved in ( [START_REF] Cohen | CLT for random walks of commuting endomorphisms on compact abelian groups[END_REF]).

Theorem 5.5. If (Z n ) is a 2-dimensional reduced centered random walk with a nite moment of order 2 and f is in AC 0 (T ρ ) with spectral density ϕ f and a non zero asymptotic variance, then for a.e. ω the FCLT holds for

is a nite set of characters on T ρ , χ 0 the trivial character.

Let 0 = t 0 < t 1 < ... < t r = 1. To use Proposition 3.2, we check the asymptotic orthogonality condition (37) and the bound (38) on cumulants:

-the variance is of order n ln n and (37) follows from (25) in Lemma 1.6, -for (38), we have to show that, for a.e. ω, C r (

We apply Theorem 3.1. Let us check (34). For r xed, the function (n 1 , ..., n r ) → m f (n 1 , ..., n r ) :=

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 (33).

Therefore, since mixing of all orders implies

1b) For f ∈ AC 0 (T ρ ), using Proposition 5.1 and Lemma 3.3, the convergence follows by approximation of f by trigonometric polynomials f L in such a way that lim L ϕ f -f L (0) = 0.

2) Moment of order 4 and tightness

Let us consider real centered functions f on T ρ in AC 0 (T ρ ), i.e., in real form such that

Taking into account Remark 3.6, it suces to consider for f a character and show that the bound is uniform, independent of the character.

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

Let α u,j , u = 1, ..., ρ , be the set of distinct eigenvalues of A j , j = 1, 2. We write α u for α

, where v 0 is some non zero component of v in a suitable basis in which A 1 , A 2 have a simultaneous triangular representation and α u = α u 1 α u 2 , with α u 1 (resp. α u 2 ) an eigenvalue of A 1 (resp. A 2 ).

The previous number is less than

where above in H we count the number of solutions without vanishing proper sub-sums.

The total ergodicity of the action implies that, if α

Therefore, for G we nd the number of self-intersections of the random walk starting from b:

Let us consider the second quantity H. Up to a permutation of indices, we can assume that i 4 < i 3 < i 2 < i 1 . We may write up to a constant factor:

By Theorem 5.3, the set of triples 1 , 2 , 3 ∈ Z 2 (without vanishing proper sub-sum) solving the equation α 1 u -α 2 u + α 3 u = 1 is a nite set F .

We can now apply [START_REF] Leonov | On the central limit theorem for ergodic endomorphisms of compact commutative groups (Russian)[END_REF] in Lemma 1.2: there exists a positive integrable function C 3 such that

where W n (ω, 1 , 2 , 3 ) = #{1 ≤ i 0 , i 1 , i 2 , i 3 ≤ n : Remark that the bounds do not depend on the character, but only on A 1 , A 2 .

The tightness property follows now from Proposition 3.5 with γ = 5. Observe that Condition (45) in this proposition is satised, since here the r.v.s are bounded.


Alexandre Benoit
email: alexandre.benoit@univ-savoie.fr

David Alleysson

Jeanny Herault

Patrick Le Callet

Spatio-Temporal Tone Mapping Operator based on a Retina model

Keywords: High Dynamic Range compression, tone mapping, retina model, color constancy, video sequence tone mapping

From moonlight to bright sun shine, real world visual scenes contain a very wide range of luminance; they are said to be High Dynamic Range (HDR). Our visual system is well adapted to explore and analyze such a variable visual content. It is now possible to acquire such HDR contents with digital cameras; however it is not possible to render them all on standard displays, which have only Low Dynamic Range (LDR) capabilities. This rendering usually generates bad exposure or loss of information. It is necessary to develop locally adaptive Tone Mapping Operators (TMO) to compress a HDR content to a LDR one and keep as much information as possible. The human retina is known to perform such a task to overcome the limited range of values which can be coded by neurons. The purpose of this paper is to present a TMO inspired from the retina properties. The presented biological model allows reliable dynamic range compression with natural color constancy properties. Moreover, its non-separable spatio-temporal filter enhances HDR video content processing with an added temporal constancy.

Introduction

In this paper, we propose a method to compress High Dynamic Range images in order to make visual data perceptible on display media with lower dynamic range capabilities. HDR images are our real life visual world; our eyes perceive everyday a wide variety of visual scenes with really different luminance values. Our visual system is able to cope with such a wide variety of input signals and extract salient information. However, we can notice that, as discussed in [START_REF] Hérault | Modeling Visual Perception for Image Processing[END_REF], neurons cannot code a wide range of input values. Thus, at the retina level a compression process occurs in order to preserve all relevant information in the coding process, including color. A similar idea has been recently introduced with digital High Dynamic Range imaging. It is now possible to create HDR images even with standard digital cameras [START_REF] Ward | High Dynamic Range Imaging: Acquisition, Display, and Image Based Lighting[END_REF] or light simulation [START_REF] Ward | The RADIANCE Lighting Simulation and Rendering System[END_REF], nevertheless the development of HDR displays, which would be able to render all the acquired data is still under development [START_REF] Hoefflinger | High-Dynamic-Range (HDR) VisionMicroelectronics, Image Processing[END_REF]. Current displays are Low Dynamic Range and direct HDR image visualization would hide a large part of the information. An alternative is the development of Tone Mapping Operators [START_REF] Ahmet | Perceptual Evaluation of Tone Reproduction Operators Using the Cornsweet-Craik-O'Brien Illusion[END_REF][START_REF] Jiangtao | Evaluating HDR rendering algorithms[END_REF] which allow HDR images to be rendered on standard LDR displays and preserve most of the information to be seen. Nevertheless, as discussed in [START_REF] Johnson | Cares and concerns of CIE TC8-08 : Spatial appearance modeling and HDR rendering[END_REF] a HDR image cannot be perceived similarly to its LDR version. Human factors related to this problem are not already known but some artifacts created by the tone mapping conversion can already be measured [START_REF] Tunç | Dynamic Range Independent Image Quality Assessment[END_REF]. They are related to halo effects and color distortions and lead to naturalness corruption. Nevertheless, studies on the quality perception [START_REF] Ahmet | Perceptual Evaluation of Tone Reproduction Operators Using the Cornsweet-Craik-O'Brien Illusion[END_REF][START_REF] Jiangtao | Evaluating HDR rendering algorithms[END_REF][START_REF] Kuang | Evaluating HDR rendering algorithms[END_REF] are the only current assessment solutions. The current challenge is to design a TMO able to limit artifacts and preserve the general ambiance of the original HDR visual scene. Several operators have already been proposed and compared [START_REF] Jiangtao | Evaluating HDR rendering algorithms[END_REF][START_REF] Kuang | Evaluating HDR rendering algorithms[END_REF] and lead to a wide variety of approaches. From computer vision methods to the one inspired by the visual system, each TMO presents a different approach and requires specific parameters set for each processed image. In addition, a new challenge is related to video sequence processing and has not yet been explored. The aim is to generate successive tone mapped images which allow a natural perception sensation without temporal instabilities created by frame-by-frame image optimization. In this paper, we propose a new TMO based on a retina model. The approach models retina local adaptation properties, as described in Meylan et al. work [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF] and is completed by specific spatio-temporal filters of the retina. The added contribution involves retina processes which enable spatio-temporal noise removal, temporal stability introduction and spectral whitening. The paper is presented as follows: section 2 describes the proposed retina model and its properties for HDR compression. Section 3 illustrates the effect of such a filter in the case of static and dynamic content processing.

Retina biological model

The human retina architecture is based on cellular layers which process the visual information from the photoreceptors visual data entry point to the ganglion cells output. The input signals are locally processed step by step so that details, motion and color information are enhanced and conditioned for high level analysis at the visual cortex level. Here, we focus on the known parts of the human retina, which are suitable for a Tone Mapping Operator design. The aim is to show that tone mapping is already performed in low level vision so that higher level visual tasks are facilitated. Furthermore, modeling these early vision properties leads to a fast and efficient TMO. We choose to work with the model described in [START_REF] Hérault | Modeling Visual Perception for Image Processing[END_REF], which takes into account the different low level processes occurring in the retina. We particularly focus on the foveal vision area and its output called Parvocellular channel, which brings the details and color information to the central nervous system. The aspects taken into account are detailed in the following:

• Local luminance and local contrast adaptation at the photoreceptor and ganglion cells levels. This biological property is directly linked to our dynamic range compression topic. • The spatio-temporal filtering occurring at the Outer Plexiform Layer level (OPL).

This filtering allows input image frequency spectrum to be whiten and enhances image details. Moreover, its temporal properties allow noise reduction and temporal stability. • Color sampling: the input image is spatially sampled by sensors with different color sensitivities. Our TMO allows gray scale and color images to be processed in the same way and introduces color constancy properties.

The architecture of the proposed model follows the biological model architecture and is depicted on figure 1. The input image can be either a raw gray image or a multiplexed color one. Then, color processing appears as an additional processing, which consists of preliminarily multiplexed color information followed by a filtering stage, then a demultiplexing stage. The key-point of the model is actually the two local adaptation steps, corresponding respectively to photoreceptors and ganglion cells, and the OPL filter placed between them. As discussed in [START_REF] Hérault | Modeling Visual Perception for Image Processing[END_REF], the photoreceptor's local adaptation is modulated by the OPL filter.

Local adaptation

Photoreceptors are able to adjust their sensitivity with respect to the luminance of their spatio-temporal neighborhood. This is modeled by the Michaelis-Menten [START_REF] Hérault | Modeling Visual Perception for Image Processing[END_REF] relation, which is normalized for a luminance range of [0, Vmax] (Eq. 1). Vmax represents the maximum pixel value in the image-255 in the case of standard 8 bits images. It may vary greatly in case of High Dynamic Range images.

(

) 1
(2)

In this relation, the response C(p) of photoreceptor p depends on the current excitation R(p) and on a compression parameter R0 (p) which is linearly linked to the local luminance L(p) (cf. Eq.2) of the neighborhood of the photoreceptor p. This local luminance L(p) is computed by applying a spatial low pass filter to the input image. This low pass filtering is actually achieved by the horizontal cells network as presented later on. Moreover, in order to increase flexibility and make the system more accurate, we add in R0 (p) the contribution of a static compression parameter V0 of value range [0;1]. Compression effect is reinforced when V0 tends to 1 or is attenuated when reaching 0. Photoreceptors V0 value is set to 0.7 as a generic experimental value.

Figure 2 shows the evolution of sensitivity with respect to R0 (p) and illustrates the effect of such a compression on a back-lit picture. Sensitivity is reinforced for low values of R0 (p) and is kept linear for high values. As a result, this model enhances contrast visibility in dark areas while maintaining it in bright areas.

OPL: Spatio-temporal filtering and contour enhancement

The cellular interactions of the OPL layer can be modeled with a non-separable spatio-temporal filter [START_REF] Hérault | Modeling Visual Perception for Image Processing[END_REF] whose transfer function for 1D signal is defined in eq. ( 3) where fs and ft denote respectively spatial and temporal frequency. Its transfer function is drawn on figure 3.a. This filter can be considered as a difference between two low-pass spatio-temporal filters which model the photoreceptor network ph and the horizontal cell network h of the retina. As discussed in [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF][START_REF] Hérault | Modeling Visual Perception for Image Processing[END_REF], the output of the horizontal cells network (Fh) is very low spatial frequency limited and can be interpreted as the local luminance L(p) required by the photoreceptors local adaptation step (eq. 2). Moreover, Fh filter's temporal low pass effect allows local luminance computation to be temporally smoothed. Finally, as a general rule, global F OPL filter has a spatio-temporal high-pass effect in low frequencies which results in a spectral whitening of the input. Its high-frequency low-pass effect enables the removal of the structural noise.

F OPL ( f s , f t ) = F ph ( f s , f t ) ⋅ 1 -F h ( f s , f t ) ( ) with F i ( f s , f t ) = 1 1+ β i + 2α i 1 -cos(2π f s ) ( )+ j2πτ i f t (3)
with subscript i=ph or i=h.

Fig. 3: OPL filter transfer function and illustration of its effect on an image and its spectrum (input image is the output of the photoreceptors previous stage).

βph is the gain of filter Fph. Setting βph to 0 cancels luminance information and a higher value allows luminance to be partially processed; βph is typically set to 0.7 which allows a good general effect. Parameters α i and τ i stand respectively for the space and time constants of the filters. Temporal noise is minimized when τph is increased. Besides, the higher τh the slower temporal adaptation. αph spatial filtering constant should be remained low (close to 0) in order to preserve high spatial frequencies. αh allows local luminance to be extracted; its generic value is set to a space constant of 7 pixels. One of the most relevant effects is spectral whitening, which compensates for the 1/f spectrum of natural images as shown in figure 3.b-c. It happens on a limited portion of the spatial frequency spectrum.

Retina ganglion cells final dynamic compression step

The ganglion cells of the Parvocellular channel receive the information coming from the OPL. They act as local contrast enhancers and are modeled by the Michaelis-Menten law similar to the photoreceptors as discussed in [START_REF] Smirnakis | Adaptation of Retinal Processing to Image Contrast and Spatial Scale[END_REF] but with different parameters. Indeed, the local luminance value is related to a smaller area around each cell since receptor fields are smaller. Also, as a general rule, the compression effect is more powerful at the ganglion cells level. The parameter V 0 is then higher than the one of photoreceptors and is typically set to 0.9. Figure 4 shows results of this local adaptation on the OPL's output stage. This last filtering allows the final luminance compression of originally very dark areas.

Color processing

To deal with color, it is possible to take advantage of the photoreceptors color sampling properties. As described in [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF], since in the fovea photoreceptors (mainly cones) sample the visual scene with three different sensitivities (Long, Medium and Short wavelengths: with L, M and S cones), the spectrum of the color sampled image presents special properties. Luminance spectrum is centered on the low frequencies while color information spectrum is located in higher frequencies. Modeling such properties consists in multiplexing the color image before processing it with the previously presented retina model and demultiplexing it afterwards, as presented in figure 1. As a consequence, designing a TMO with such an approach while preserving high spatial frequencies allows luminance map to be toned and color information to be kept. Moreover, from a computational cost point of view, processing color only requires an additional color multiplexing/demultiplexing stage to the gray level processing core. Towards this end, we propose to use the color demultiplexing algorithm presented in [START_REF] Chaix De Lavarène | Practical Implementation of LMMSE Demosaicing Using Luminance and Chrominance Spaces[END_REF] which supports different color sampling methods (Bayer, Diagonal, Random).

Following the TMO scheme depicted on figure 1, figure 5 shows the color management steps of the algorithm. Using Bayer color sampling, color spectrum is translated in high spatial frequencies. Thus, since OPL filter with a low value of αph keeps those frequencies, color information remains unchanged.

To sum up, the human retina naturally achieves color tone mapping and its simplified model allows digital HDR images to be processed in the same way. Notice that this current application is suitable to deal with high dynamic range in terms of luminance; issues related to gamut limitation are not considered.

Design of a retina based TMO

As shown in the previous section, the retina model includes many properties required for low level image tone mapping. Indeed, its preliminary goal is to condition visual signals in order to make them compliant with the subsequent image analysis "operators". In addition to its image processing properties, this model has the ability to present a low computational cost: the gray level processing part requires only 16 operations per pixel. Color management depends on the chosen demultiplexing method -here it requires less than 200 linear operation per pixels. Consequently, we follow the biological model and use all the tools described on figure 1. Actually, the proposed TMO is rather similar to the one proposed by Meylan & al. [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF] but it adds the contribution of the OPL filter which achieves a more complete retina model, taking into account temporal aspects. The main advantage is that the OPL intermediate filter allows spectral whitening and temporal information management. For simplification, we first describe its properties for static visual scenes and then deal with video sequences processing.

Static content

Even if human factors related to HDR imaging are not well known, some critical points have been reported in literature. A brief overview of the main issues and the response introduced by our method are presented in the following. Refer to figure 6 for image samples.

• The first point is related to artifacts generation in the tone mapping process: since luminance is compressed, algorithms have a tendency to generate halos and contours around the boundaries of high luminance gradient of visual scenes [START_REF] Ahmet | Perceptual Evaluation of Tone Reproduction Operators Using the Cornsweet-Craik-O'Brien Illusion[END_REF][START_REF] Mantiuk | Display Adaptive Tone Mapping[END_REF]; e.g, sun and lights boundaries. Our algorithm limits such artifacts since low spatial frequency energy is lowered and logarithmic compression of photoreceptors is very local (typically 7-pixel wide). • Color constancy is the second challenge. It is often impaired when color components are manipulated independently [START_REF] Ward | High Dynamic Range Imaging: Acquisition, Display, and Image Based Lighting[END_REF]. Due to multiplexing stage, our method naturally remains color information in high spatial frequencies and do not change them. Furthermore, objects color information is preserved, independently of the illuminant color [START_REF] Hérault | Modeling Visual Perception for Image Processing[END_REF]. • Natural rendering is the general issue for image tone mapping. Even though details extraction in bright and dark areas of a visual scene is the final goal, it can change the image initial balance and reduce its dynamic [START_REF] Ward | High Dynamic Range Imaging: Acquisition, Display, and Image Based Lighting[END_REF][START_REF] Yoshida | Perceptual Evaluation of Tone Mapping Operators with Real-World Sceness[END_REF][START_REF] Jiangtao | Evaluating HDR rendering algorithms[END_REF]. Global ambiance luminance should be preserved in order to maintain the initial « goal of the image » and the areas at which the photographer wants the observers to look at. Concerning that specific issue, our algorithm allows luminance ambiance to be less compressed by lowering the value of photoreceptors V 0 parameter. It is a matter of trade-off: either luminance ambiance survival or all details extraction. In addition, it is generally difficult to identify a unique TMO parameters setting which would allow many different images to be tone mapped and look "natural". Each TMO algorithm generally has its own parameters which should be optimized for each image, resulting in a supervised tone mapping process. However, some algorithms already present good parameters stability and generate natural-looking images with a large set of pictures, e.g. the Meylan et al. [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF], Mantiuk's [START_REF] Mantiuk | A Perceptual Framework for Contrast Processing of High Dynamic Range Images (revised and extended version)[END_REF] algorithm and our contribution.

Figure 6 gives examples of tone mapping results with the proposed algorithm and other results coming from [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF], another human visual system based TMO [START_REF] Mantiuk | A Perceptual Framework for Contrast Processing of High Dynamic Range Images (revised and extended version)[END_REF], which includes visual cortex models but less retina properties, and Reinhard's TMO [START_REF] Ward | High Dynamic Range Imaging: Acquisition, Display, and Image Based Lighting[END_REF], which is more focused on local adaptation retina properties. Note that generic default values were used as unique parameter set for each method. On these examples, color constancy is achieved for both [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF] and our method since they manage color in a similar way. Meylan's method compresses luminance to a larger extent, which allows details to be more visible. In parallel, the proposed method keeps better global luminance ambiance and provide minimal halo effects. As a compromise, dark details may remain hidden because the variance of R0 (p) is less due to an additional temporal filtering. While comparing all methods, we can see how different results can be from one algorithm to the other. Actually, apart from constancy problems, naturalness and artifacts limitation versus details extraction trade off is the most difficult to reach. Also, our method, [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF] and [START_REF] Mantiuk | A Perceptual Framework for Contrast Processing of High Dynamic Range Images (revised and extended version)[END_REF] appear in these samples as optimal TMOs with a constant parameters set. However, only future visual quality tests will allow rigorous comparisons to be made. Fig. 6 : tone mapping results comparison between our approach and other existing methods (refer to [16] for higher resolution results). Our algorithm does not generate halos (refer to the two last examples), nevertheless, dark areas' details can be hidden (refer to the first two examples). The other compared algorithms can better extract details in dark areas but halos or color distortion can appear. Also, locked parameters and accurate tone mapping are not supported by all the algorithms presented in this sample.

Video content

As mentioned in [START_REF] Didyk | Enhancement of Bright Video Features for HDR Displays[END_REF], TMO encounter critical problems when visual content changes with time. Indeed, from frame-to-frame critical light changes can appear and since current TMO are independently optimized for each frame, the reconstructed video sequence can be degraded by successive uncorrelated tone mapping operations. This problem is even more difficult to overcome when dealing with global image TMO, which generate a tone mapped image depending on the global image luminance. This problem is more limited in the case of local operators such as [START_REF] Meylan | A Model of Retinal Local Adaptation for the Tone Mapping of Color Filter Array Images[END_REF][START_REF] Mantiuk | Display Adaptive Tone Mapping[END_REF] and our method. However, video content tone mapping is a very recent challenge and it is not already possible to compare results due to the lack of real HDR video contents. Here, we propose in figure 7 some samples of a HDR video sequence shoot to illustrate some properties of our method. Sudden light changes do not impact global luminance of the tone mapped images. Fig. 7: HDR video sequence tone mapping results samples (refer to [16] for higher resolution results). In this test sequence, the sun generates optic halos on the camera, the most important changes are related to sudden sun disappearing and appearing depending on camera position and objects on the visual scene. Our method allows details to be constantly extracted whatever luminance change there are. Moreover, colors and image dynamic remain stable.

Our TMO presents temporal constants, which acts on local luminance processing adaptation. Particularly, the spatio-temporal horizontal cells low pass filter of the OPL filter computes the local luminance with a memory effect. This results in smooth transitions when luminance changes suddenly. Also, the photoreceptors stage allows spatio-temporal noise removal at the beginning of the tone mapping process.

Conclusion

This paper proposes a Tone Mapping Operator (TMO) based on a human retina model. It imitates some parts of the foveal retina functionalities including its luminance compression properties and adds the contribution of temporal information processing. This "biologic" algorithm is able to process a wide variety of images with natural rendering and has a good potential for video sequences processing. Moreover, the method presents good color constancy properties and its color management flexibility enables easier practical implementation. Future work could aim at identifying human factors, which would allow to identify the main clues of the most appropriate TMO and adjust its parameters in that sense, as well as visual perception tests in order to further assess the subjective quality of the Tone Mapping operators.

Fig. 1 :

 1 Fig. 1: simplified model of the proposed retina model. Color processing is an additional processing, which requires preliminary input image multiplexing and output picture demultiplexing.

Fig. 2 :

 2 Fig. 2: photoreceptors local luminance adaptation. Left: output response with regard to the local R0 (p) value. Right: effect on a HDR image (from: www.openexr.org).

Fig. :

 : Fig.: Effect of the ganglion cells local adaptation (input image is the output of the OPL previous stage)

Fig. 5 :

 5 Fig. 5: color multiplexing and demultiplexing on the TMO "boundaries".

Acknowledgement

This work was supported by FuturIm@ge project within the "Media and Network" French Cluster. Also, we thank Grzegorz Krawczyk from MPI Informatik who gave us access to these rare HDR video sequences.