Françoise Pène

LIMIT THEOREMS FOR ADDITIVE FUNCTIONALS OF RANDOM WALKS IN RANDOM SCENERY

Keywords: Brownian motion, ergodicity, innite measure, dynamical system

We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process. These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n 1 4 ). When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n 1 8 ). 2000 wthemtis ujet glssi(tionF 60F05; 60F17; 60G15; 60G18; 60K37.

1. Introduction 1.1. Description of the model and of some earlier results. We consider two independent sequences (X k ) k≥1 (the increments of the random walk) and (ξ y ) y∈Z (the random scenery) of independent identically distributed Z-valued random variables. We assume in this paper that X 1 is centered and admits nite moments of all orders, and that its support generates the group Z. We dene the random walk (S n ) n≥0 as follows S 0 := 0 and S n := n i=1 X i for all n ≥ 1.

We assume that ξ 0 is centered, that its support generates the group Z, and that it admits a nite second moment σ 2 ξ := E[ξ 2 0 ] > 0. The random walk in random scenery (RWRS) is the process dened as follows

Z n := n-1 k=0 ξ S k = y∈Z ξ y N n (y) , (1)
where we set N n (y) = #{k = 0, . . . , n -1 : S k = y} for the local time of S at position y before time n. This process rst studied by Borodin [START_REF] Borodin | e limit theorem for sums of independent rndom vriles de(ned on reurrent rndom wlkF (Russian)[END_REF] and Kesten and Spitzer [START_REF] Kesten | e limit theorem relted to new lss of selfEsimilr proessesF Z[END_REF] describes the evolution of the total amount won until time n by a particle moving with respect to the random walk S, starting with a null amount at time 0 and wining the amount ξ at each time the particle hits the position ∈ Z. This process is a natural example of (strongly) stationary process with long time dependence. Due to the rst works by Borodin [START_REF] Borodin | e limit theorem for sums of independent rndom vriles de(ned on reurrent rndom wlkF (Russian)[END_REF] and by Kesten and Spitzer [START_REF] Kesten | e limit theorem relted to new lss of selfEsimilr proessesF Z[END_REF], we know that (n -3 4 Z nt ) t converges in distribution, as n goes to innity, to the so-called Kesten and Spitzer process (σ ξ ∆ t , t ≥ 0), where ∆ is dened by

∆ t := +∞ -∞ L t (x) dβ x , (2)
with (β x ) x∈R a Brownian motion and (L t (x), t ≥ 0, x ∈ R) a jointly continuous in t and x version of the local time process of a standard Brownian motion (B t ) t≥0 , where ((B t ) t , (β s ) s ) is the limit in distribution of n -1 2 ((S nt ) t , (σ -1 ξ ns k=1 ξ k ) s ) as n → +∞. Observe that ∆ is the continuous time analog of the random walk in random scenery. To be convinced of this fact, one may compare the right hand side of (1) with [START_REF] Aurzada | ersistene proilities for sttionry inrement proesses[END_REF]. The process ∆ is a classical and nice example of a (strongly) stationary process, self-similar with dependent (strongly) stationary increments and exhibiting long time dependence.

In [START_REF] Borodin | e limit theorem for sums of independent rndom vriles de(ned on reurrent rndom wlkF (Russian)[END_REF], Borodin established the convergence in distribution of Z n when X and ξ have second order moments. Kesten and Spitzer established in [START_REF] Kesten | e limit theorem relted to new lss of selfEsimilr proessesF Z[END_REF] a functional limit theorem when the distributions of X and ξ belong to the domain of attraction of stable distributions with respective parameters α = 1 and β ∈ (0, 2]. Limit theorems have been extended by Bolthausen [START_REF] Bolthausen | entrl limit theorem for twoEdimensionl rndom wlks in rndom seneriesF[END_REF] (for the case α = β = 2 for random walks of dimension d = 2), by Deligiannidis and Utev [START_REF] Deligiannidis | esymptoti vrine of the selfEintersetions of stle rndom wlks using hrouxE iener theory[END_REF] (α = d ∈ {1, 2}, β = 2, providing some correction to [START_REF] Bolthausen | entrl limit theorem for twoEdimensionl rndom wlks in rndom seneriesF[END_REF]) and by Castell, Guillotin-Plantard and the author [START_REF] Castell | vimit theorems for one nd twoEdimensionl rndom wlks in rndom seneryF Ann I[END_REF] (when α ≤ d and β < 2), completing the picture for the convergence in the sense of distribution and for the functional limit theorem (except in the case α ≤ 1 and β = 1). Since the seminal works by Borodin and by Kesten and Spitzer, random walks in random scenery and the Kesten and Spitzer process ∆ have been the object of various studies (let us mention for example [START_REF] Khoshnevisan | he odimension of the zeros of stle proess in rndom seneryF Séminaire de Probabilités XXXVII[END_REF][START_REF] Xiao | he rusdor' dimension of the level sets of stle proesses in rndom seneryF[END_REF][START_REF] Guillotin-Plantard | uenhed entrl limit theorems for rndom wlks in rndom seneryF Stochastic Process[END_REF][START_REF] Berger | heteting the tril of rndom wlker in rndom seneryF Electron[END_REF][START_REF] Guillotin-Plantard | enewl theorems for rndom wlks in rndom seneryF Electron[END_REF][START_REF] Guillotin-Plantard | he quenhed limiting distriutions of oneEdimensionl rnE dom wlk in rndom seneryF Electron[END_REF][START_REF] Guillotin-Plantard | impiril proesses for reurrent nd trnsient rndom wlks in rndom senery[END_REF][START_REF] Aurzada | ersistene proilities for sttionry inrement proesses[END_REF]).

Random walks in random scenery are related to other models, such as the Matheron and de Marsily Model [START_REF] Matheron | ss trnsport in porous medi lwys di'usivec e ounterxmple[END_REF] of transport in porous media, the transience of which has been established by Campanino and Petritis [START_REF] Campanino | ndom wlks on rndomly oriented ltties[END_REF] and which has many generalizations (e.g. [START_REF] Guillotin-Plantard | rnsient rndom wlks on PdEoriented ltties[END_REF][START_REF] Devulder | ndom wlk in rndom environment in twoEdimensionl strti(ed medium with orienttions[END_REF][START_REF] Gantert | yn the reurrene of some rndom wlks in rndom environment[END_REF][START_REF] Brémont | yn plnr rndom wlks in environments invrint y horizontl trnsltions[END_REF][START_REF] Brémont | lnr rndom wlk in strti(ed qusiEperiodi environment[END_REF]), and such as the Lorentz-Lévy process (see [START_REF] Pène | ndom wlks in rndom seneries nd relted models[END_REF] for a short presentation of some models linked to random walks in random scenery).

Random walks in random scenery constitute also a model of interest in the context of dynamical systems. They correspond indeed to Birkho sums of a transformation called the T, T -1 transformation appearing in [49, p. 682, Problem 2] where it was asked whether this Kolmogorov automorphism is Bernoulli or not. In [START_REF] Kalikow | EI rnsformtion is xot voosely fernoulli[END_REF], Kalikow answered negatively this question by proving that this transformation is not even loosely Bernoulli. 1.2. Main results. Before stating our main results, let us introduce some additional notations. Let d ∈ N be the greatest common divisor of the set {x ∈ Z, P(ξ 0 -ξ 1 = x) > 0} and α ∈ Z such that P(ξ 0 = α) > 0. This means that the random variables ξ take almost surely their values in α + dZ and that d is largest positive integer satisfying this property. Since the support of ξ generates the group Z, necessarily α and d are coprime. Recall that the quantity d can be also simply characterized using the common characteristic function ϕ ξ of the ξ . 1 In the present paper we are interested in the asymptotic behaviour of additive functionals of the RWRS (Z n ) n≥1 that is of quantities of the following form:

Z n := n k=1 f (Z k )
where f : Z → R is absolutely summable. This quantity is strongly related to the local time N n of the RWRS Z, which is dened by N n (a) = #{k = 1, ..., n : Z k = a} .

Indeed if f = 1 0 , then Z n = N n (0) and if f = 1 0 -1 1 , then Z n = N n (0) -N n [START_REF] Aaronson | en introdution to in(nite ergodi theory[END_REF]. In the general case, Z n can be rewritten The asymptotic behaviour of (N n (0)) n has been studied by Castell, Guillotin-Plantard, Schapira and the author in [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Corollary 6], in which it has been proved that the moments of (n -1 4 N n (0)) n≥1 converge to those of the local time L 1 (0) at position 0 and until time 1 of the process ∆. The proof of this result was based on a multitime local limit theorem [14, Theorem 5] extending a local limit theorem contained in [START_REF] Castell | e lol limit theorem for rndom wlks in rndom senery nd on rndomly oriented lttiesF[END_REF] and on the niteness of the moments of L 1 (0) (which was a delicate question). We complete this previous work by establishing in Section 2 the following bounds for the moments of L 1 (0). Theorem 1. For any η 0 > 0, there exists a > 0 and C > 0 such that

(Cm) 3m 4 ≤ E[(L 1 (0)) m ] = O a m (m!) 3 2 +η 0 Γ( m 4 + 1)
≤ O m m( 5 4 +2η 0) .

Even if it uses some ideas that already existed in [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF], the proof of Theorem 1 (given in Section 2) is dierent in many aspects. It requires indeed much more precise estimates which changes in the approach of the control of the moments. The proof of Theorem 1 relies on several auxiliary results. We summarize quickly its strategy. We will prove (see [START_REF] Brémont | lnr rndom wlk in strti(ed qusiEperiodi environment[END_REF] coming from [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF] and [START_REF] Brémont | yn plnr rndom wlks in environments invrint y horizontl trnsltions[END_REF]) that

E[(L 1 (0)) m ] = m! (2πσ 2 ξ ) m 2 0<t 1 <...<tm<1 m-1 k=0 (t k+1 -t k ) -3 4 E m-1 k=0 d(L (k+1) , W k ) -1 dt 1 ...dt m ,
where we set W k := V ect(L (1) , ..., L (k) ) and L (k+1) := (L t k+1 -L t k )/(t k+1 -t k )

3 4 (normalized so that |L (m) | L 2 (R) has the same distribution as |L 1 | L 2 (R) ). We will prove, in Lemma 7, that

∃c, C > 0, m! 0<t 1 <...<tm<1 m-1 k=0 (t k+1 -t k ) -3 4 dt 1 ...dt m ∼ c(Cm) 3m 4 ,
as m → +∞ and, in Lemma 6, that

E |L 1 | -1 L 2 (R) m ≤ E m-1 k=0 d(L (k+1) , W k ) -1 ≤ m-1 k=0 sup V ∈V k E (d (L 1 , V )) -1 , (3)
where d(•, •) is the distance associated with the L 2 -norm on L 2 (R) and where V k is the set of linear subspaces of L 2 (R) of dimension at most k. Theorem 1 will then follow from the next self-interesting estimate on the local time L 1 of the Brownian motion B up to time 1.

Theorem 2.

sup

V ∈V k E (d (L 1 , V )) -1 = k 1 2 +o(1) , as k → +∞ . (4)
Now we use the following classical argument for positive random variables. The upper bound provided by Theorem 1 allows us to prove that the Carleman's criterion is satised for E L 1 (0) where E is a centered Rademacher distribution independent of L 1 (0) and of Z, indeed:

m≥1 E[(L 1 (0)) m ] -1 2m ≥ c 1 m≥1 m -5 8 -η 0 = ∞ ,
for every η 0 ∈ (0, 3 8 ). This enables us to deduce from [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Corollary 6] that n -

1 8 E N n (0)
converges in distribution to E σ -1 ξ L 1 (0) and so that

n -1 4 N n (0) L -→ σ -1 ξ L 1 (0) , as n → +∞ , (5)
where L -→ means convergence in distribution. This convergence in distribution is extended to more general observables.

Theorem 3. Let f : Z → R be such that a∈Z |f (a)| < ∞. Then n -1 4 n-1 k=0 f (Z k ) converges in distribution and in the sense of moments to a∈Z f (a)σ -1 ξ L 1 (0).
The proof of the moments convergence in Theorem 3 is a straigthforward adaptation of [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF] and is given in Appendix B. Due to Theorem 1 and to the above argument that lead to [START_REF] Blachère | e rossover for the d on(gurtions of rndom wlk in rndom seneryF[END_REF], the convergence in distribution in Theorem 3 is a consequence of the moments convergence. Another strategy to prove the convergence in distribution in Theorem 3 consists in seing this result as a direct consequence of (5) combined with Proposition 13 stating the ergodicity of the dynamical system ( Ω, T , µ) corresponding to

T k ((X m+1 ) m∈Z , (ξ m ) m∈Z , Z 0 ) = ((X k+m+1 ) m∈Z , (ξ m+S k ) m∈Z , Z k ) .
This dynamical system preserves the innite measure µ := P ⊗Z X 1 ⊗ P ⊗Z ξ 0 ⊗ λ Z , where λ Z is the counting measure on Z. Actually, thanks to [START_REF] Blachère | e rossover for the d on(gurtions of rndom wlk in rndom seneryF[END_REF] and to the recurrence ergodicity of ( Ω, T , µ), we prove the following stronger version of the convergence in distribution of Theorem 3.

Theorem 4. For any µ-integrable function f : Ω → R,

n -1 4 n-1 k=0 f • T k L( µ) -→ Ω f d µ σ ξ L 1 (0) , as n → +∞ ,
where

L( µ)
-→ means convergence in distribution with respect to any probability measure absolutely continuous with respect to µ.

Theorem 3 can be seen as weak law of large numbers, with a non constant limit. When a∈Z f (a) = 0, the limit given by Theorem 3 vanishes, but then the next result provides a limit theorem for Z n = n-1 k=0 f (Z k ) with another normalization. This second result corresponds to a central limit theorem for additive functionals of RWRS. Let us indicate that, contrarily to the moments convergence in Theorem 3, the next result is not an easy adaptation of [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF], even if its proof (given in Section 4) uses the same initial idea (computation of moments using the local limit theorem) and, at the begining, some estimates established in [START_REF] Castell | e lol limit theorem for rndom wlks in rndom senery nd on rndomly oriented lttiesF[END_REF][START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]. Indeed, important technical diculties arise from the cancellations coming from the fact that a∈Z f (a) = 0. Theorem 5. Assume moreover that there exists some κ ∈ (0, 1] such that ξ 0 admits a moment of order 2 + κ.

Let f : Z → R be such that a∈Z (1 + |a|)|f (a)| < ∞ and that a∈Z f (a) = 0. Then ∈Z d-1 =0 a,b∈Z 2 f (a)f (b)P(Z | +d | = a -b) < ∞ .

Moreover all the moments of

n -1 8 n-1 k=0 f (Z k ) n converges to those of σ 2 f σ ξ L 1 (0)N ,
where N is a standard Gaussian random variable independent of L 1 (0) and where

σ 2 f := k∈Z a,b∈Z 2 f (a)f (b)P(Z |k| = a -b) . (6)
In particular, for any a ∈ Z, n -1 8 (N n (a) -N n (0)) n converges in the sense of moments to

σ 2 0,a σ ξ L 1 (0)N , with σ 2 0,a := k∈Z 2P(Z |k| = 0) -P(Z |k| = a) -P(Z |k| = -a) .
Let us point out the similarity beween these results and the classical Law of Large Numbers and Central Limit Theorem for sums of square integrable independent and identically distributed random variables. Indeed Theorems 3 and 5 establish convergence results of the respective following forms

1 a n n k=1 Y k → I(Y 1 )Y and 1 √ a n n k=1 (Y k -I(Y 1 )Y 0 k ) → σ 2 Y Y Z
as n → +∞, with a n → +∞, I an integral (with respect to the counting measure on Z) and Y 0 k a reference random variable with integral 1 (e.g.

Y 0 k = 1 0 (Z k ), note that we cannot take Y 0 k = 1
since it is not integrable with respect to the counting measure on Z).

The summation order in the expression (6) of σ 2 f is important. Indeed recall that P(Z k = 0) has order k -3 4 and so is not summable. The sum k∈Z appearing in ( 6) is a priori non absolutely convergent if d = 1. Indeed, considering for example that ξ 0 is a centered Rademacher random variable (i.e. P(ξ 0 = 1) = P(ξ 0 = -1) = 1 2 ) and that

f = 1 0 -1 1 , then, for any k ≥ 0, a,b∈Z 2 f (a)f (b)P(Z |2k| = a -b) = P(Z |2k| = 0 -0) + P(Z |2k| = 1 -1) = 2P(Z |2k| = 0) and a,b∈Z 2 f (a)f (b)P(Z |2k+1| = a -b) = -P(Z |2k+1| = 0 -1) -P(Z |2k+1| = 1 -0) = -P(|Z |2k+1| | = 1) .
But, σ 2 f corresponds to the following sum of an absolutely convergent series (in k):

σ 2 f = k∈Z   d-1 =0 a,b∈Z 2 f (a)f (b)P(Z | +dk| = a -b)   .
Finally, let us point out that σ 2 f dened in (6) corresponds to the Green-Kubo formula, wellknown to appear in central limit theorems for probability preserving dynamical systems (see Remark 14 at the end of Section 3).

Let us indicate that results similar to Theorem 5 exist for one-dimensional random walks, that is when the RWRS (Z n ) n≥1 is replaced by the RW (S n ) n≥1 , with other normalizations and with an exponential random variable instead of L 1 (0). Such results have been obtained by Dobru²in [START_REF] Dobru²in | wo limit theorems for the simplest rndom wlk on lineF (Russian) Uspehi Mat[END_REF], Kesten in [START_REF] Kesten | yuption times for wrkov nd semiEwrkov hinsF Trans[END_REF] and by Csáki and Földes in [START_REF] Csáki | yn symptoti independene nd prtil sums, Asymptotic methods in probability and statistics[END_REF][START_REF] Csáki | esymptoti independene nd dditive funtionls[END_REF]. The idea used therein was to construct a coupling using the fact that the times between successive return times of (S n ) n≥1 to 0 are i.i.d., as well as the partial sum of the f (S k ) between these return times to 0 and that these random variables have regularly varying tail distributions. This idea has been adapted to dynamical contexts by Thomine [START_REF] Thomine | héorèmes limites pour les sommes de firkho' de fontions d9intégrle nulle en théorie ergodique en mesure in[END_REF][START_REF] Thomine | e generlized entrl limit theorem in in(nite ergodi theoryF Probab[END_REF]. Still in dynamical contexts, another approach based on moments has been developed in [START_REF] Thomine | otentil kernelD hitting proilities nd distriutionl symptotisF Ergodic Theory and Dynamical Systems[END_REF][START_REF] Thomine | gentrl limit theorems for the ¢ PEperiodi vorentz gs[END_REF] in parallel to the coupling method. This second method based on local limit theorem is well tailored to treat non-markovian situations, such as RWRS. Indeed, recall that the RWRS (Z n ) n≥1 is (strongly) stationary but far to be not markovian (for example it has been proved in [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF] that Z n+m -Z n is more likely to be 0 if we know that Z n = 0) and even more intricate conditionally to the scenery (it has been proved in [START_REF] Guillotin-Plantard | he quenhed limiting distriutions of oneEdimensionl rnE dom wlk in rndom seneryF Electron[END_REF] that the RWRS does not converge knowing the scenery). Luckily local limit theorem type estimates enables to prove moments convergence. But unfortunately Theorem 1 is not enough to conclude the convergence in distribution via Carleman's criterion.

The paper is organized as follows. In Section 2, we prove Theorem 1 (bounds on moments of the local time of the Kesten Spitzer process) and Theorem 2 (estimate on the distance in L 2 (R) between the local time of a Brownian motion and a k-dimensional vector space). In Section 3, we establish the recurrence ergodicity of the innite measure preserving dynamical system ( Ω, T , µ) and obtain the convergence in distribution of Theorem 3 (Law of Large Numbers) as a byproduct of this recurrence ergodicity combined with [START_REF] Blachère | e rossover for the d on(gurtions of rndom wlk in rndom seneryF[END_REF]. Section 3 is completed by Appendix B which contains the proof of the moments convergence of Theorem 3. In Section 4 (completed with Appendix A), we prove Theorem 5 (Central Limit Theorem).

Upper bound for moments: Proof of Theorem 1

This section is devoted to the study of the behaviour of E[(L 1 (0)) m ] as m → +∞. It has been proved in [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF] that this quantity is nite, but the estimate established therein was not enough to apply the Carleman criterion. The proof of Theorem 1 requires a much more delicate study, even if it uses some estimates used in [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]. We start by establishing bounds for

E[(L 1 (0)) m ]. Lemma 6. E |L 1 | -1 L 2 (R) m m! (2πσ 2 ξ ) m 2 0<t 1 <...<tm<1 m-1 k=0 (t k+1 -t k ) -3 4 dt 1 ...dt m ≤ E[(L 1 (0)) m ] (7)
and

E[(L 1 (0)) m ] ≤ m-1 j=0 sup V ∈V k E (d (L 1 , V )) -1 m! (2πσ 2 ξ ) m 2 0<t 1 <...<tm<1 m-1 k=0 (t k+1 -t k ) -3 4 dt 1 ...dt m , (8)
where d(f, g) = |f -g| L 2 (R) and where V k is the set of linear subspaces of L 2 (R) of dimension at most k.

Proof. Recall that it has been proved in [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Theorem 3] that where d(f, g) = f -g L 2 (R) and where V ect(L t 1 , ..., L tm ) is the sublinear space of L 2 (R) generated by L t 1 , ..., L tm . It follows that

E[(L 1 (0)) m ] = m! (2πσ 2 ξ ) m 2 0<t 1 <...<tm<1 E[(det D t 1 ,...,tm ) -1 2 ] dt 1 ...dt m , (9) with
det D -1 2 t 1 ,...,tm = m-1 k=0 d(L t k+1 , V ect(L t 1 , ..., L t k )) -1 . (10)
But, for any m ≥ 1 and any 0 < t 1 < ... < t m+1 < 1 and any k = 0, ..., m -1,

E d L t k+1 , V ect(L t 1 , ..., L t k ) -1 (B s ) s≤t k = E d L t k+1 -L t k , V ect(L t 1 , ..., L t k ) -1 (B s ) s≤t k = E d (L t k+1 -L t k )(B t k + •), V ect(L t 1 (B t k + •), ..., L t k (B t k + •)) -1 (B s ) s≤t k .
Therefore

E L t k+1 -L t k -1 L 2 (R) ≤ E d L t k+1 , V ect(L t 1 , ..., L t k ) -1 (B s ) s≤t k (11)
and

E d L t k+1 , V ect(L t 1 , ..., L t k ) -1 (B s ) s≤t k ≤ sup V ∈V k E d (L t k+1 -L t k )(B t k + •), V -1 , ( 12
)
where V k is the set of linear subspaces of dimension at most k of L 2 (R) and where we used the independence of (L t k+1 -L t k )(B t k + •) with respect to (B s ) s≤t k and the fact that

(L t 1 (B t k + •), ..., L t k (B t k + •)
) is measurable with respect to (B s ) s≤t k . Thus, by induction and using the fact that the increments of B are (strongly) stationary, it follows from [START_REF] Brémont | yn plnr rndom wlks in environments invrint y horizontl trnsltions[END_REF] and ( 12) that

m-1 k=0 E L t k+1 -L t k -1 L 2 (R) ≤ E det D -1 2 t 1 ,...,tm ≤ m-1 k=0 sup V ∈V k E d (L t k+1 -L t k )(B t k + •), V -1 = m-1 k=0 sup V ∈V k E d L t k+1 -t k , V -1 , (13)
with the convention t 0 = 0. Recall that (L u (x)) x∈R has the same distribution (

√ uL 1 (x/ √ u)) x∈R
and so (d(L u , V ect(g 1 , ..., g k ))) 2 has the same distribution as

min a 1 ,...,a k R √ uL 1 x √ u - k i=1 a i g i (x) 2 dx = u min a 1 ,...,a k R L 1 x √ u - k i=1 a i g i (x) 2 dx = u 3 2 min a 1 ,...,a k R L 1 (y) - k i=1 a i g i ( √ uy) 2 dy = u 3 2 (d(L 1 , V ect(h 1 , ..., h k ))) 2
(t k+1 -t k ) -3 4 E |L 1 | -1 L 2 (R) ≤ E det D -1 2 t 1 ,...,tm ≤ m-1 k=0 (t k+1 -t k ) -3 4 sup V ∈V k E (d (L 1 , V )) -1 , (14)
which ends the proof of the lemma.

We rst study the behaviour, as m → +∞, of the integral appearing in Lemma 6.

Lemma 7. m! 0<t 1 <...<tm<1 m-1 k=0 (t k+1 -t k ) -3 4 dt 1 ...dt m = m! Γ( 1 4 ) m Γ( m 4 + 1) ∼ c(Cm) 3m 4 ,
as m → +∞.

Proof.

a m+1 := 0<t 1 <...<t m+1 <1 m k=0 (t k+1 -t k ) -3 4 dt 1 ...dt m+1 = x i >0 : x 1 +...+x m+1 <1 m+1 k=1 x -3 4 k dx 1 ...dx m+1 = 1 0 x -3 4 m+1 (1 -x m+1 ) -3m 4 x i >0 : x 1 +...+xm<1-x m+1 m k=1 (x k /(1 -x m+1 )) -3 4 dx 1 ...dx m dx m+1 = 1 0 x -3 4 m+1 (1 -x m+1 ) m 4 u i >0 : u 1 +...+um<1 m k=1 u -3 4 k du 1 ...du m dx m+1 = a m 1 0 x -3 4 m+1 (1 -x m+1 ) m 4 dx m+1 = a m B 1 4 , m 4 + 1 = a m Γ( 1 4 )Γ( m 4 + 1) Γ( m+1 4 + 1)
, where B(•, •) and Γ stand respectively for Euler's Beta and Gamma functions, and so, by induction,

a m = Γ(1/4) m Γ( m 4
+1) proving the rst point of the lemma. Moreover

m!a m ∼ (Γ(1/4)) m m m+ 1 2 (m + 4) -m 4 -1 2 4 m 4 + 1 2 e -3m 4 +1 ,
where we used the Stirling formulas m! = Γ(m + 1) and

Γ(z) ∼ √ 2πz z-1 Observe that E |L 1 | -1 L 2 (R) > 0.
Thus, the proof of Theorem 1 will be be deduced from the two previous lemmas combined with Theorem 2, which can be rewritten as follows

∀η 0 > 0, ∃C > 1, ∀k ∈ N * , C -1 k 1 2 -η 0 ≤ sup V ∈V k E (d (L 1 , V ))) -1 ≤ Ck 1 2 +η 0 . ( 15
)
Due to [START_REF] Revuz | gontinuous mrtingles nd frownin motion[END_REF]Cor. (1.8) of Chap. VI, Theorem (2.1) of Chap. I], L 1 is almost surely Hölder continuous of order 1 2 -η 0 and its Hölder constant admits moments of any order. The lower bound of theorem 2 follows directly from this fact.

Proof of the lower bound of Theorem 2. We prove the lower bound of [START_REF] Chen | ndom lk sntersetionsX vrge hevitions nd elted opisF Mathematical Surveys and Monographs[END_REF]. Let η 0 ∈ (0, 1 2 ). Let C 1 be the Hölder constant of order 1 2 -η 0 of L 1 . Let V k be the linear subspace of L2 (R) generated by the set

1 [m/k,(m+1)/k] , m = - k 2 , ..., k 2 -1 ,
and consider L k ∈ V k given by

L k := k 2 -1 m=-k 2 L 1 m k 1 [ m k , m+1 k ) .
Let K 0 > 0. We will use the fact that

E (d (L 1 , V k )) -1 ≥ E (d (L 1 , V k )) -1 1 {C 1 ≤K 0 , sup [0,1] |B|≤ k-1 2k } . Observe that, if sup [0,1] |B| ≥ k-1 2k and C 1 ≤ K 0 , then d (L 1 , V k ) 2 ≤ d(L 1 , L k ) 2 = k 2 -1 m= k 2 m+1 k m k (L 1 (u) -L 1 (m/k)) 2 du ≤ k 2 -1 m= k 2 k -1 K 0 k -1 2 +η 0 2 ≤ K 0 k -1 2 +η 0 2 . Thus E (d (L 1 , V k )) -1 ≥ E (d (L 1 , V k )) -1 1 {C 1 ≤K 0 , sup [0,1] |B|≤ k-1 2k } ≥ E K 0 k -1 2 +η 0 -1 1 {C 1 ≤K 0 , sup [0,1] |B|≤ k-1 2k } ≥ K -1 0 k 1 2 -η 0 P C 1 ≤ K 0 , sup [0,1] |B| ≤ 1 3 .
The rest of this section is devoted to the proof of the upper bound of Theorem 2 (i.e. the upper bound of ( 15)), which is much more delicate to establish. To this end, we will prove a sequence of estimates. Let us rst introduce the quantities used in this proof. We x η 0 > 0 and 1 10 ) such that

d = 1 2 + η 0 > 1/2. Choose 0 ∈ (0,
d > 1 + 0 2(1 -0 ) . (16)
Fix a, b, η, γ ∈ (0, 1 10 ) such that 0 < b 8 < a 2 and small enough so that

(1 + γ)(1 + 0 ) 2 + a 2 + b 8 < 1 (17)
and

(2d(1 -0 ) -1 -0 )(1 -2η) -8η > 0 . ( 18
) Let θ > 0 such that (1 -2η)θ > 1 and 1 - b 4 - (1 + γ)(1 + 0 ) 2 < θ(1 -2η) 1 - (1 + γ)(1 + 0 ) 2 - a 2 - b 8 (19)
and

(1 -0 )(1 + 2d) < θ [(2d (1 -0 ) -1 -0 )(1 -2η) -8η] . (20)
The existence of such a θ is ensured by ( 17) and [START_REF] Csáki | esymptoti independene nd dditive funtionls[END_REF]. Fix then K such that 1 4a-b < K and v 0 = 16/b . We will also consider the following quantities which will depend on k ≥ 1. We set M := θk and M := M d . For x > M , we also set:

r 0 := (x/M ) -(1+γ)(1+ 0 ) M -1+ 0 2 M -1-0 , x 0 = (x/M ) a M, x 1 = (x/M ) b . (21)
Let V be a linear space generated by g 1 , ..., g k ∈ L 2 (R). Observe that

E (d (L 1 , V ))) -1 = ∞ 0 P (d (L 1 , V ))) -1 > x dx = O(M ) + ∞ M P d (L 1 , V )) < x -1 dx . ( 22
)
Lemma 8. Uniformly on x > M :

P d (L 1 , V )) < x -1 ≤ O (x/M ) -2 + P ∀ = -v 0 , ..., v 0 , D L 1 x -1 8 1 + n x 0 n=1,...,M , W ( x -1 8 1 ) V < 2x -1 r -1 2 0 . ( 23
)
where

W (y 0 ) V := Span y 0 +(n+1)/x 0 y 0 +n/x 0 g j (y ) dy n=1,...,M , 1, ..., k ⊂ R M and where D is the usual euclidean metric in R M .
Proof. We set

C 1 := sup y,z∈R : y =z |L 1 (y) -L 1 (z)| |y -z| u , with u := 1 1 + 0 - 1 2 .
Since C 1 admits moments of every order, it follows that

P (d (L 1 , V ) < 1/x) ≤ P d (L 1 , V )) < 1/x, C 1 ≤ (x/M ) γ + O((x/M ) -2 ) ,
Note that, if x > M , then

r 0 x 0 = (x/M ) a-(1+γ)(1+ 0 ) M 1-0 2 M -1-0 ≤ 1 , as soon as x > M , since a < 1 < (1+γ)(1+ 0 ) and since M = M d with 1 2 ≤ d, and so r 0 ≤ x -1 0 . Assume moreover that d(L 1 , V ) < 1/x and C 1 ≤ x γ . Let a j such that d L 1 , k j=1 a j g j < x -1 .
Then, for every ∈ Z, the following estimate holds true

x -1 >   M n=1 x -1 8 1 + n x 0 +r 0 x -1 8 1 + n x 0   L 1 (y) - k j=1 a j g j (y)   2 dy   1 2 ≥   M n=1 x -1 8 1 + n x 0 +r 0 x -1 8 1 + n x 0   L 1 x -1 8 1 + n x 0 - k j=1 a j g j (y)   2 dy   1 2 -M r 0 (x/M ) 2γ r 2u 0 1 2 ≥ √ r 0 D L 1 x -1 8 1 + n x 0 n=1,...,M , W ( x -1 8
1 ) V - √ M (x/M ) γ r 1 2 +u 0 . Since 1 2 +u = 1 1+ 0 and r 0 = (x/M ) -(1+γ)(1+ 0 ) M -1+ 0 2 M -1-0 , we conclude that √ M (x/M ) γ r 1 2 +u 0 =
x -1 and so

P (d (L 1 , V )) < 1/x, C 1 ≤ x γ ) ≤ P ∀ = -v 0 , ..., v 0 , D L 1 x -1 8 1 + n x 0 n , W ( x -1 8
1 ) V < 2x -1 r -1 2 0 . Recall that v 0 = 16/b . Set E 0,W :=    D   x 0 n+1 x 0 n x 0 Y (y) dy n=1,...,M , W   < 2x -1 √ x 0    . Lemma 9.
The following estimate holds true uniformly on x > M :

P d (L 1 , V )) < x -1 ≤ O (x/M ) -2 + sup W P (E 0,W ) , (24)
where sup W means the supremum over the set of linear subspaces W of R M of dimension at most k and where Y is a squared Bessel process of dimension 0 starting from x Proof. We adapt the proof of [14, Lemma 9].

Setting := x -1 8
1 and T u := min{s > 0 : |B s | = u} for the rst hitting time of {±u} by the Brownian motion B, we observe that

P(T v 0 > 1) = P sup s∈[0,1] |B s | ≤ v 0 = O(e -c 0 (v 0 ) -2 ) = O (x/M ) -bv 0 /8 = O (x/M ) -2 .
(25) (using e.g. [43, Proposition 8.4, page 52]). Moreover, due to [44, Exercise 4.12, Chapter VI, p 265], for every n = 0, ..., v 0 -1,

P L T (n+1) (B T n ) -L T n (B T n ) ≤ ( ) 2 |(B u ) u≤T n ≤ P(L T (0) ≤ ( ) 2 ) ≤
and so, due to the strong Markov property,

P ∀n = 0, ..., v 0 -1, L T (n+1) (B T n ) -L T n (B T n ) ≤ ( ) 2 ≤ ( ) v 0 ,
and this, combined with [START_REF] Guillotin-Plantard | he quenhed limiting distriutions of oneEdimensionl rnE dom wlk in rndom seneryF Electron[END_REF], ensures that there exists

C 0 > 0 such that P(∀ = -v 0 , ..., v 0 , L 1 ( ) ≤ ( ) 2 ) ≤ C 0 ( ) v 0 and so P (∀ = -v 0 , ..., v 0 , t (x 1 ) > 1) ≤ C 0 (x/M ) -bv 0 /8 ≤ C 0 (x/M ) -2 , ( 26
)
setting t (x 1 ) := inf{s > 0 : L s ( x -1 8 1 ) > x -1 4 1 }.
Moreover, for any = 1, ..., v 0 , we have

sup V P t (x 1 ) < 1, D L 1 x -1 8 1 + n x 0 n , W ( x -1 8
1 ) V < 2x -1 √ x 0 ≤ sup V P t (x 1 ) < 1, D L t (x 1 ) x -1 8 1 + n x 0 n , W ( x -1 8
1 ) V,0 < 2x -1 √ x 0 ≤ sup W P D L t (x 1 ) x -1 8 1 + n x 0 n , W < 2x -1 √ x 0 , (27)
setting

W ( x -1 8 1 ) V,0 := W ( x -1 8 1 ) V + (L t (x 1 ) -L 1 ) x -1 8 1 + n x 0 n . Due to the second Ray-Knight theorem (see [44, Theorem 2.3, page 456]), (L t (x 1 ) ( x -1 8
1 + y)) y≥0 has the same distribution as Y . The lemma follows from ( 26) and [START_REF] Guillotin-Plantard | enewl theorems for rndom wlks in rndom seneryF Electron[END_REF].

Recall that 4a -b > 0. Set E 1 := sup s≤M/x 0 |Y (s) -x -1/4 1 | < x -1/4 1 2 .
Lemma 10. For every K > (4a -b) -1 , the following estimate holds true uniformly on x > M :

P (E 1 ) = 1 -O (x/M ) -K(4a-b) . (28)
Proof. Using the Burkholder-Davis-Gundy inequality, combined with the fact that Y is dominated by the square of a Brownian motion starting from x

-1 4
1 , we observe that

p x = P sup s≤M/x 0 |Y (s) -x -1/4 1 | ≥ x -1/4 1 2 ≤ C K x 2K 1 2 8K E   M/x 0 0 Y (u) du 4K   ≤ C K x 2K 1 (M/x 0 ) 4K-1 M/x 0 0 E Y (u) 4K du (29)
with C K = 2 8K C K , and so

p x ≤ C K x 2K 1 (M/x 0 ) 4K-1 M/x 0 0 E (x -1/8 1 + B u ) 8K du ≤ C K x 2K 1 (M/x 0 ) 4K 2 8K x -K 1 + (M/x 0 ) 4K ≤ C K x K 1 (M/x 0 ) 4K + x 2K 1 (M/x 0 ) 8K with C K = 2 8K C K and x K 1 (M/x 0 ) 4K = (x/M ) -K(4a-b) , since x 0 = (x/M ) a M and x 1 = (x/M ) b . Lemma 11. Uniformly on x > M , sup W P (E 0,W ∩ E 1 ) ≤C k (x/M ) 1-b 4 - (1+γ)(1+ 0 ) 2 k-M (1-2η) 1- (1+γ)(1+ 0 ) 2 -a 2 -b 8 × M (1-0 )k 4 + (1+ 0 )(1-2η)M 4 +2ηM M 1-0 2 (k-(1-2η)M ) .
Proof. Observe that

E 0,W ∩ E 1 ⊂ Y (n/x 0 ) n=1,...,M ∈ B ∞ x -1/4 1 , x -1/4 1 2 ∩ W x , (30)
where

B ∞ x -1/4 1 , x -1/4 1 2
is the ball (for the supremum norm) of radius x -1/4 1 2 and centered on

(x -1/4 1 , ..., x -1/4 1
), and where

W x is the ε = 2x -1 r -1 2 0 -neighbourhood of W for the metric D. Note that = 2(x/M ) (1+γ)(1+ 0 ) 2 -1 M 1+ 0 4 M 1+ 0 2 -1 (31)
and

R x := √ M x -1/4 1 ε = 1 2 (x/M ) 1-b 4 - (1+γ)(1+ 0 ) 2 M 1-0 4 M 1- 1+ 0 2 1 , (32)
uniformly in x > M , since b 4 + (1+γ)(1+ 0 ) 2 < 1. Observe that B ∞ x -1/4 1 , x -1/4 1 2 ∩W x is contained in B 2 x -1/4 1 , √ M x -1/4 1 ∩ W x where B 2 x -1/4 1 , √ M x -1/4 1
is the euclidean ball centered on

(x -1/4 1 , ..., x -1/4 1
) with radius

√ M x -1/4 1 . Let z 0 , z 0 ∈ B 2 x -1/4 1 , √ M x -1/4 1 ∩ W x and z 1 ∈ W ∩ B 2 (z 0 , ε), z 1 ∈ W ∩ B 2 (z 0 , ε). Then z 1 ∈ B 2 z 1 , 3 √ M x -1/4 1
. Due to [45, Theorem 3, pages 157], there exists c > 0 such that

W ∩ B 2 z 1 , 3 √ M x -1/4 1 is contained in the union of at most (cR x ) k euclidean balls of radius ε in W . Thus W x ∩ B 2 x -1 4 1 , √ M x -1/4 1 is contained in the union of at most (cR x ) k euclidean balls of radius 2ε. We conclude that B ∞ x -1/4 1 , x -1/4 1 2 ∩W
x is contained in the union of at most

(cR x ) k euclidean balls of radius 4ε centered at a point contained in B ∞ x -1/4 1 , x -1/4 1 2 ∩ W x .
It follows from this combined with [START_REF] Kalikow | EI rnsformtion is xot voosely fernoulli[END_REF] that

P [E 0,W ∩ E 1 ] ≤ (cR x ) k sup z∈B∞ x -1/4 1 , x -1/4 1 2 P (Y (n/x 0 )) n=1,...,M ∈ B 2 (z, 4ε) . ( 33
) Note that if z ∈ B ∞ x -1/4 1 , x -1/4 1 2 and (Y (n/x 0 )) n=1,...,M ∈ B 2 (z, 4ε), then max n=0,...,M -1 |z n+1 - x -1 4 1 | < x -1 4 1
2 and there exist at most ηM indices n that |Y (n /x 0 ) -z n | ≥ 4ε/ √ ηM , and so at least

(1 -2η)M indices n = {0, ..., M -1} such that Y n x 0 -z n , Y n + 1 x 0 -z n+1 < 4ε/ ηM , with z 0 = x -1 4
1 . Due to [44, after Corollary 1.4, page 441], the distribution of Y ((n + 1)/x 0 ) knowing Y (n/x 0 ) = y is the sum of a Dirac mass at 0 and of a measure with density

z → q x 0 (y, z) := x 0 2 y z exp - x 0 (y + z) 2 I 1 (x 0 √ yz) ,
where I 1 is the modied Bessel function of index 1 which satises I 1 (z) = O(e z / √ z), as z → ∞, (see [35, (5.10.22) or (5.11.10)]). So

q x 0 (y, z) = O x 1 2 0 x 1 8 1 exp - x 0 ( √ y - √ z) 2 2 = O x 1 2 0 x 1 8 1 uniformly on y, z ∈ x -1 4 1 4 , 2x -1 4 1
. We will use the expression x 0 , x 1 and given in [START_REF] Dobru²in | wo limit theorems for the simplest rndom wlk on lineF (Russian) Uspehi Mat[END_REF] and [START_REF] Kesten | yuption times for wrkov nd semiEwrkov hinsF Trans[END_REF].

Thus by using the Markov property (and

M ! (M (1-2η))!(2ηM )! ≤ M 2ηM ), we get by induction, that, when x > M , sup z∈B∞ x -1/4 1 , x -1/4 1 2 P (Y (n/x 0 )) n=1,...,M ∈ B 2 (z, 4ε) ≤ M 2ηM C (x/M ) -1- (1+γ)(1+ 0 ) 2 -a 2 -b 8 M 1+ 0 4 M -1+ 1+ 0 2 (1-2η)M
.

Recalling that M = O(k), the previous estimate combined with [START_REF] Khoshnevisan | he odimension of the zeros of stle proess in rndom seneryF Séminaire de Probabilités XXXVII[END_REF] and [START_REF] Kesten | e limit theorem relted to new lss of selfEsimilr proessesF Z[END_REF] ensures that

sup W P (E 0,W ∩ E 1 ) ≤C k (x/M ) 1-b 4 - (1+γ)(1+ 0 ) 2 k-M (1-2η) 1- (1+γ)(1+ 0 ) 2 -a 2 -b 8 M (1-0 )k 4 + (1+ 0 )(1-2η)M 4 +2ηM M 1- 1+ 0 2 (k-(1-2η)M ) , (34)
which ends the proof of the lemma.

Proof of the upper bound of Theorem 2. Formula (15) follows from [START_REF] Dombry | hisrete pproximtion of stle selfEsimilr sttionry inrements proessF[END_REF] and Lemmas 9, 10 and 11. We will use the fact that

∀Q > 1, ∞ M (x/M ) -Q dx = O(M ) . (35)
Thanks to this, the error terms in Lemmas 9 and 10 gives directly a term in

O(M ) = O(k d ).
Let us detail the term coming from Lemma 11. We rst observe that the exponent of (x/M ) is strictly smaller than -1 for k large enough. Indeed this exponent is

1 - b 4 - (1 + γ)(1 + 0 ) 2 k -M (1 -2η) 1 - (1 + γ)(1 + 0 ) 2 - a 2 - b 8
which is smaller than

k 1 - b 4 - (1 + γ)(1 + 0 ) 2 -θ(1 -2η) 1 - (1 + γ)(1 + 0 ) 2 - a 2 - b 8
where we used the fact that M = θk ≥ θk. The fact that this quantity is strictly smaller than -1 for any k large enough comes from our conditions ( 17) and [START_REF] Deligiannidis | esymptoti vrine of the selfEintersetions of stle rndom wlks using hrouxE iener theory[END_REF]. It follows from this combined with [START_REF] Lebedev | peil funtions nd their pplitions[END_REF] and Lemma 11 that

+∞ M sup W P (E 0,W ∩ E 1 ∩ E 2 ) dx ≤ C k M (1-0 )k 4 + (1+ 0 )(1-2η)M 4 +2ηM M 1+ 1-0 2 (k-(1-2η)M ) ≤ C k M d+ (1-0 )(1+2d)M 4θ + (1+ 0 -2d(1-0 ))(1-2η)M 4 +2ηM ,
where we used the fact that M = M d and that k ≤ θk /θ = M/θ. Finally, we notice that 1 + 0 -2d(1 -0 ) < 0 (due to ( 16)) and that [START_REF] Devulder | ndom wlk in rndom environment in twoEdimensionl strti(ed medium with orienttions[END_REF] ensures that

(1 -0 )(1 + 2d) 4θ + (1 + 0 -2d(1 -0 ))(1 -2η) 4 + 2η < 0 and conclude that +∞ M sup W P (E 0,W ∩ E 1 ∩ E 2 ) dx = O(1) .

Law of large numbers: Proof of Theorem 3

We complete the sequence (X n ) n≥1 into a bi-innite sequence (X n ) n∈Z of i.i.d. random variables. Theorem 3 could be proved by an adaptation of the proof of [14, Corollary 6] (combined with Theorem 1). We use here another approach enabling the study of more general additive functionals. Recall that (ξ m+S k ) m∈Z is the scenery seen from the particle at time k.

Proposition 12. Let f : Z Z × Z Z × Z → R be a measurable function such that ∈Z |E[ f ((X n+1 ) n∈Z , (ξ n ) n∈Z , )]| < ∞ . Then n-1 k=0 f ((X m+k+1 ) m∈Z , (ξ m+S k ) m∈Z , Z k+m ) N n (0) n≥0 converges almost surely to I( f ) := ∈Z E[ f ((X n ) n∈Z , (ξ n ) n∈Z , )].
In particular, this combined with (5) ensures that

n -1 4 n-1 k=0 f ((X m+k+1 ) m∈Z , (ξ m+S k ) m∈Z , Z k ) n≥0 converges in distribution to I( f )σ -1 ξ L 1 (0).
Our approach to prove Proposition 12 uses an ergodic point of view. Let us consider the probability preserving dynamical system (Ω, T, µ) given by

Ω = Z Z × Z Z , T ((x k ) k∈Z , (y k ) k∈Z ) = ((x k+1 ) k∈Z , (y k+x 0 ) k∈Z ), µ = P ⊗Z X 1 ⊗ P ⊗Z ξ 0 ,
i.e. T (x, y) = (σx, σ x 0 y), where we write σ : Z Z → Z Z for the usual shift transformation given by σ ((z k ) k∈Z ) = (z k+1 ) k∈Z . This system is known to be ergodic (see [START_REF] Weiss | he isomorphism prolem in ergodi theory[END_REF][START_REF] Kalikow | EI rnsformtion is xot voosely fernoulli[END_REF]). We set Φ(x, y) := y 0 . With these notations, Z k corresponds to the Birkho sum n-1 k=0 Φ•T k . Consider the Z-extension ( Ω, T , µ) over (Ω, T, µ) with step function Φ. This system is given by

Ω := Ω × Z, µ = µ ⊗ λ Z ,
where λ Z = ∈Z δ is the counting measure on Z and with T (x, y, ) = (T (x, y), + y 0 ) .

In particular

T k ((x m+1 ) m∈Z , (y m ) m∈Z , ) =   (x m+k+1 ) m∈Z , (y m+x 0 +...+x k-1 ) m∈Z , + k-1 j=0 y x 0 +...+x j   .
Observe that N n (0) corresponds to the Birkho sum n-1 k=0 h 0 • T k (x, y, 0) with h 0 (x, y, ) = 1 0 ( ), and the sum studied in Proposition 12 corresponds to n-1 k=0 f • T k (x, y, 0), while

I( f ) = Ω f d µ.
Proposition 13. The system ( Ω, T , µ) is recurrent ergodic.

Proof. Since (Ω, T, µ) is ergodic and since Φ is integrable and µ-centered, we know (by [46, Corollary 3.9] combined with the Birkho ergodic theorem) that P(Z n = 0 i.o.) = 1, thus that ( Ω, T , µ) is recurrent (i.e. conservative). Now let us prove that this system is also ergodic. Let g : Ω → (0, +∞) be a positive µ-integrable function such that g(x, y, ) = g 0 ( ) does not depend on (x, y) ∈ Ω and with unit integral (g is a probability density function with respect to µ). By recurrence of ( Ω, T , µ), we know that

k≥1 g • T k = ∞ (36)
µ-almost everywhere. Let K ∈ N. Consider f : Ω → R a µ-integrable function constant on the K-cylinders of the rst coordinate, i.e. such that f (x, y, ) = f 0 ((x m ) |m|≤k , y, ) does not depend on (x k ) |k|>K .

Since ( Ω, T , µ) is recurrent, the Hopf-Hurewicz's theorem (see e.g. [1, p. 56]) ensures that

lim |n|→+∞ n k=1 f • T k n k=1 g • T k = H (f,g) := E g µ f g I (37)
µ-almost everywhere, where I is the σ-algebra of T -invariant events. Thus the ergodicity of ( Ω, T , µ) will follow from the fact that H (f,g) is µ-almost everywhere constant for every f as above (g can be xed). Observe that, for k > K,

f • T k (x, y, ) = f σ k x, σ x 0 +...+x k-1 y, + k-1 m=0 y x 0 +...+xm = f 0 x k-K , ..., x K+k , σ x 0 +...+x k-1 y, + k-1 m=0 y x 0 +...+xm does not depend on (x k ) k≤-1 . Analogously, for k > K, f • T -k (x, y, ) = f σ -k x, σ -x -1 -...-x -k y, - k m=1 y -x -1 -...-x -m = f 0 x -K-k , ..., x -(k-K) , σ -x -1 -...-x -k y, - k m=1 y -x -1 -...-x -m
does not depend on (x k ) k≥0 . Of course g • T k satises the same property. Thus, due to ( 36) and [START_REF] Gall | wouvement rownienD proessus de rnhement et superproessus[END_REF], it follows that H(f, g)(x, y, ) does not depend on x. Thus,

H (f,g) (x, y, ) = H (0) (f,g) (y, )
for µ-almost every (x, y, ) ∈ Ω.

By T -invariance of H (f,g) , given two distinct points x 0 , x 0 ∈ Z such that P(X 1 = x 0 )P(X 1 = x 0 ) > 0, the following equality holds true almost everywhere

H (0) (f,g) (y, ) = H (0) (f,g) (σ x 0 y, + y 0 ) = H (0) (f,g) (σ x 0 y, + y 0 ) ,
where we write σ for the usual shift on Z Z given by σ((y k ) k∈Z ) = (y k+1 ) k∈Z . It follows that, for every ∈ Z, H (0) (f,g) (•, ) is σ x 0 -x 0 -invariant almost everywhere. By ergodicity of σ x 0 -x 0 , we conclude that H (f,g) (x, y, ) = H (1) f,g ( ) depends only on almost everywhere. Since it is T -invariant, for every y 0 ∈ Z such that P(ξ 0 = y 0 ) > 0, H (1) f,g ( ) = H

(1) f,g ( + y 0 ). Since the support of y 0 generates the group Z, we conclude that H (f,g) is µ-almost everywhere equal to a constant. Note that the system in innite measure ( Ω, T , µ) describes the evolution in time m of ((X m+k+1 ) k∈Z , (ξ Sm+k ) k , Z m ). In comparison, the system corresponding to ((X m+k+1 ) k , S m ) is also recurent ergodic, but the analogous system corresponding to ((X m+k+1 ) k , (ξ Sm+k ) k , S m ) is recurrent (since P(S n = 0 i.o.) = 1) not ergodic (since the sets of the form {(x, y, ) : (y n-) n ∈ A 0 } are invariant).

Proof of Proposition 12. Since ( Ω, T , µ) is recurrent ergodic, the Hopf ergodic theorem ensures that, for any f ∈ L 1 ( µ), the sequence

n-1 k=0 f • T k n-1 k=0 h 0 • T k n≥0
converges µ-almost everywhere to We end this section with an interpretation of σ 2 f in terms of the famous Green-Kubo formula.

Ω f d µ Ω h 0 d µ = I( f ). Thus n-1 k=0 f ((X m+k+1 ) m∈Z , (ξ m+S k ) m∈Z , Z k+m ) N n (0) = n-1 k=0 f • T k n-1 k=0 h 0 • T k ((X m ) m∈Z , (ξ m ) m∈Z , 0)
Remark 14. Assume the assumptions of Theorem 5. consider the function f : Ω → Z given by f (x, y, ) := f ( ). Then σ 2 f can be rewritten

σ 2 f = k∈Z Ω f . f • T |k| d µ .
4. Proof of the central limit theorem: proof of Theorem 5

We start by stating key intermediate results. We recall that d and α have been introduced in the beginning of Section 1.2. Proposition 15. Assume the assumptions of Theorem 5. Let M ∈ N * and η > 0. There exists L ∈ (0, 1) such that for every θ ∈ (0, 1) the following holds true with the notations n j := k j -k j-1 , with the convention k 0 = 0.

First,

k j =0,...,d-1, ∀j∈J E   m j=1 f (Z k j +k j ) s j s=1 f (Z k j + j,s )   = O m i=1 n -3 4 i E k , (38)
uniformly over the k = (k 1 , ..., k m ) and = ( j,s ) j=1,...,m;s=1,...,s j such that n > k j > k j-1 + n θ (with convention k 0 := 0) and j,s ∈ {0, ..., n Lθ } with M = m j=1 (s j + 1), where we set J := {j = 1, ..., m : s j = 0} and k j = 0 if j ∈ J , and with

E k = O   J ⊂{1,...,m} : #J ≥#J /2   j∈J n -1 2 +η j     . Second, if s j = 1 for all j, then E   m j=1 f (Z k j )f (Z k j + j )   = d m E k (2πσ 2 ξ ) m 2 m j=1 A k j , j + O   n -L(M +1)θ m j=1 n -3 4 j   ,
uniformly on k, as above, with E k depending on k but not on and such that

E k = O m j=1 n -3 4 j
uniformly on k as above, and

E k ∼ n -3m 4 E det D -1 2 t 1 ,.
..,tm as k j /n → t j and n → +∞, with

D t 1 ,...,tm = ( R L t i (x)L t j (x) dx) i,j=1,.
..,m where L is the local time of the brownian motion B, limit of (S nt / √ n) t as n goes to innity, and where

A k, := a∈kα+dZ, b∈Z f (a) m s=1 f (b) P(Z = b -a) . Third, d-1 k 1 ,...,k m =0 n κθη 10M 1 ,..., m=0 2 #{j: j >0} m j=1 A k j +k j , j = σ 2m f + o(1) ,
as (k 1 /n, ..., k m /n) → (t 1 , ..., t m ) and n → +∞.

Proof. The proof of Proposition 15 is based on several technical lemmas. For reader convenience, the most technical points are proved in Appendix A. Let M ≥ 1, θ ∈ (0, 1) and η ∈ 0, 1 100 . Choose L = κη 10M . Assume n θ < n j < n and let j,1 , ..., j,s j = 0, ..., n Lθ with m j=1 (1+s j ) = M . We set N j (y) := #{s = 0, ..., n j -1 : S k j-1 +s = y}, N * j := sup y N j and R j := #{y ∈ Z : N j (y) > 0}. Analogously, we set N j,s = #{s = 0, ..., j,s -1 : S k j +s = y}. The left hand side of (38) can be written

B k, = a,b   m j=1 f (a j ) s j s=1 f (b j,s )   p k, (a, b) , ( 39
)
where a,b means the sum over (a, b) ∈ Z M with a = (a 1 , ..., a m ) and b = (b j,s ) j=1,...,m;s=1,...,s j , with the convention a 0 = 0 and p k, (a, b) = P(∀j = 1, ..., m, Z k j = a j , ∀s = 1, ..., s j , Z k j + j,s = b j,s ) .

An classical computation (detailed in Appendix A) ensures the following. Lemma 16.

p k, (a, b) = 1 {∀i, a i =k i α+dZ} d m (2π) M [-π d , π d ] m ×[-π,π] M -m e -i m j=1 [(a j -a j-1 )θ j + s j s=1 (b j,s -a j )θ j,s ] ϕ k, (θ, θ ) d(θ, θ ) . (40)
with θ = (θ j ) j=1,...,m and θ = (θ j,s ) j=1,...,m;s=1,...,s j and

ϕ k, (θ, θ ) = E   y∈Z ϕ ξ   m j=1 θ j N j (y) + s j s=1 θ j,s N j,s (y))     . (41)
For any event E and any

I ⊂ [-π d , π d ] m × [-π, π] M -m , we also set ϕ k, (θ, θ , E) = E   1 E y∈Z ϕ ξ   m j=1 θ j N j (y) + s j s=1 θ j,s N j,s (y))     , (42)
p k, (a, b, I, E) = 1 {∀i, a i =k i α+dZ} d m (2π) M I e -i m j=1 [(a j -a j-1 )θ j + s j s=1 (b j,s -a j )θ j,s ] ϕ k, (θ, θ , E ) d(θ, θ ) .
(43) and

B k, ,I,E = a,b   m j=1 f (a j ) s j s=1 f (b j,s )   p k, (a, b, I, E) . ( 44
) Let γ < min(Lθ, ηθ 2M ). Let θ ∈ (0, θη 2 ) such that θ ≤ θ 2 -2M Lθ.
We consider the set

Ω k := det D k ≥ n -θ m i=1 n 3 2 i ∩ m j=1 Ω (j) k , (45)
with

Ω (j) k :=    sup r=0,...,n j |S r+k j-1 -S k j-1 | ≤ n 1 2 +γ j 3 , sup y =z |N j (y) -N j (z)| |y -z| 1 2 ≤ n 1 4 + γ 2 j    ,
and with

D k = y∈Z N i (y)N j (y) i,j
. The following lemma follows from [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF] (see appendix A for details).

Lemma 17. For any p > 1, P(Ω k ) = 1 -o(n -p ), and so

B k, ,[-π d , π d ] M ,Ω c k = o(n -p ). Note that, on Ω k , R j ≤ n 1 2 +γ j , ( 46
)
N * j = |N * j -0| ≤ n 1 4 + γ 2 j ((n j ) 1 2 +γ ) 1 2 n 1 2 + η 2 j , (47)
V j := z∈Z (N j (z)) 2 ≥ z∈Z N j (z) 2 R j ≥ n 2 j n 1 2 +γ j ≥ n 3 2 -η 2 j , (48)
V j ≤ R j (N * j ) 2 ≤ n 3(1+η) 2 j . ( 49
)
It will be useful to notice that

ϕ k, (θ, θ , E) ≤ E   1 E y∈F ϕ ξ   m j=1 θ j N j (y)     (50)
with

F := y ∈ Z : ∀(j, s), N j,s (y) = 0 ,
and that 4 ) .

#(Z \ F) ≤ m j=1 s j s=1 j,s ≤ M n Lθ = o(n 1
(

) 51
Using a straighforward adaptation of the proof of [13, Proposition 10], we prove (see Appendix A) that Lemma 18.

B k, ,I

k ,Ω k = o e -n c , (1)
uniformly on k, as in Proposition 15, where

I (1) k is the set of (θ, θ ) ∈ -π d , π d m × [-π, π] M -m
such that there exists j = 1, ..., m so that n

-1 2 +η j < |θ j |. Lemma 19. B k, ,I (2) k ,Ω k = O   m j=1 n -5 4 +η j   , (53)
uniformly on k, as in Proposition 15, where I (2) k is the set of (θ, θ

) ∈ -π d , π d m × [-π, π] M -m
such that for all j = 1, ..., m, |θ j | < n -1 2 +η j and there exists j = 1, ..., M such that n

-1 2 -η j < |θ j |.
It remains to estimate the integral over I (3) k , the set of (θ, θ

) ∈ -π d , π d m × [-π, π] M -m such
that for all j = 1, ..., m, |θ j | < n -1 2 -η j . We set J := {j = 1, ..., m : s j = 0} = {j(1), ..., j(J)}.

Lemma 20. Under the assumptions of Theorem 5 with a∈Z f (b + ad) = 0 for all b ∈ Z. Let J ⊂ J , then

k j =0,...,d-1, ∀j∈J B k+k , ,I (3) k ,Ω k = O     m j=1 n -3 4 j   J ⊂J ∪(J +1) : #J ≥J /2   j∈J n -1 2 +η j     ,
(54) uniformly on k, as in Proposition 15, and where we set k = (k 1 , ..., k m ) with k j = 0 if j ∈ J . Moreover, if s j = 1 for all j (and J = ∅), then,

B k+k , ,I (3) k ,Ω k = d √ 2πσ ξ m a 1 ,...,am∈Z 1 {∀i, a i =k i α+dZ} E (det D k ) -1 2 1 Ω k m j=1 f (a j )E f a j + Z j + O   n -(M +1)Lθ m j=1 n -3 4 j   ,
uniformly on k, as above, with

E (det D k ) -1 2 1 Ω k = O   m j=1 n -3 4 j   ,
uniformly on k as above, and

E (det D k ) -1 2 1 Ω k ∼ n -3m 4 E det D -1 2 t 1 ,...,tm .
as k j /n → t j and n → +∞.

We can now complete the proof of Proposition 15. The two rst points of Proposition 15 comes from the upper bounds provided by Lemmas 16,[START_REF] Csáki | yn symptoti independene nd prtil sums, Asymptotic methods in probability and statistics[END_REF][START_REF] Csáki | esymptoti independene nd dditive funtionls[END_REF]19 and 20,with

E k := E (det D k ) -1 2 1 Ω k .
It remains to prove the last point of Proposition 15. We assume that s j = 1 for all j and that k j /n → t j and n → +∞. Recall that d 0 = min{n ≥ 1 : nξ 0 ∈ dZ} = min{n ≥ 1 : nα ∈ dZ}. Observe that, for every a j ∈ Z there is a unique k ∈ {0, ..., d 0 -1} such that a j ∈ (k j +k j )α+dZ. Thus

d-1 k 1 ,...,k m =0 n κθη 10M 1 ,..., m=0 2 #{j: j >0} m j=1 A k j +k j , j = n κθη 10M 1 ,..., m=0 2 #{j: j >0} a j , b j ∈Z m j=1 f (a j )f (b j )P Z j = b j -a j .
Finally, due to the last point of Lemma 20 and to the next lemma, this quantity is equivalent to

1 ,..., m≥0 2 #{j: j >0} a j , b j ∈Z m j=1 f (a j )f (b j )P Z j = b j -a j ,
as k j /n → t j and n → +∞. Proof. The proof of this lemma only uses estimates established in [START_REF] Castell | e lol limit theorem for rndom wlks in rndom senery nd on rndomly oriented lttiesF[END_REF]. Since a,b |f (a)f (b)| < ∞ and using Lemma 16, we observe that

d-1 =0 a,b∈Z f (a)f (b)P(Z + d = b -a) = d-1 =0 a∈Z b∈a+( d+ )α+dZ f (a)f (b) d 2π [-π d , π d ] e -it(b-a) E   y∈Z ϕ ξ (tN d+ (y))   dt .
Moreover, due to [13, Propositions 8,9,10], on Ω k when |t| ≤ k -3 4 +η , we have

P(Ω k ) = 1 -o(k -1-η 0 ) (due
[-π d , π d ] e -it(b-a) E   y∈Z ϕ ξ (tN d+ (y))   = |t|≤ -3 4 +η e -it(b-a) E   y∈Z ϕ ξ (tN d+ (y)) 1 Ω d+   dt + o( -1-η 0 ) = |t|≤ -3 4 +η e -it(b-a) E   y∈Z ϕ ξ (tN d (y)) 1 Ω d   dt + o( -1-η 0 ) ,
using also the fact that #{y ∈ Z : d . Now, due to (48), V -1

N d (y) = N d+ (y)} ≤ d. It follows that d-1 =0 a,b∈Z f (a)f (b)P(Z + d = b -a) = d 2π |t|≤ -3 4 +η a,b∈Z f (a)f (b) e -it(b-a) -1 E   y∈Z ϕ ξ (tN d (y)) 1 Ω d   dt + o( -1-η 0 ) ≤ d 2π |t|≤ -3 4 +η a,b |f (a)f (b)t(b -a)| E e - σ 2 ξ t 2 V d 4
1 Ω d dt + o( -1-η 0 ) ≤ CE V -1 d 1 Ω d + o( -1-η 0 ) ,
d 1 Ω d ≤ -3 2 -2γ = O( -1-η 0 )
up to take η 0 small enough, which ends the proof of the lemma. 2 Our proof is valid in a more general context. The assumptions on f and S can be relaxed in a∈Z |af (a)| < ∞, a∈Z f (a) = 0, and Sn

L 8 3 = O( √ n).
Theorem 5 follows directly from the following corollary of Proposition 15 and Lemma 21, since

E[N 2N ] = (2N )! N !2 N and E[(L 1 (0)) N ] = [0,1] N E det D -1 2 t 1 ,...,t N (2π) N 2
dt 1 ...dt N (due to [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]). Corollary 22. [A rewritting of Theorem 5]Under the assumptions of Theorem 5,

E   n k=1 f (Z k ) 2N +1   = o n 2N +1 8
, and

E   n k=1 f (Z k ) 2N   = (2N )! N !2 N n 2N 8 σ 2N f (2πσ 2 ξ ) N 2 [0,1] N E det D -1 2 t 1 ,...,t N dx 1 ...dx N .
Proof. Since f is bounded, it is enough to prove the result for n = n d. We start by writing

E   n k=1 f (Z k ) M   = 1≤m 1 ≤...≤m M ≤n c m E   M j=1 f (Z m j )   , (55)
where c m is the number of (r 1 , ..., r M ) ∈ {1, ..., n} M such that r 1 , ..., r M and m 1 , ..., m M contain the same values with same multiplicities.

Let

θ 0 ∈ 0, 1 M +1 . Given a sequence 1 ≤ m 1 ≤ ... ≤ m M ≤ n with convention m 0 = 0, we consider p ∈ {0, ..., M } such that no m j -m j-1 (for j = 1, ..., M ) is in (n L p+1 θ 0 , n L p θ 0 ]. Set θ = L p θ 0 .
We write k 1 = m 1 and, inductively, if k j = m u(j) , we set k j+1 = m u(j+1) for the smallest integer m r such that m r > k j + n θ , s j = u(j + 1) -u(j) -1 and then j,s = m u(j)+s . Thus each m = (m 1 , ..., m M ) with 1 ≤ m 1 ≤ ... ≤ m M ≤ n can be represented by at least one

(k, ) ∈ M p=0 M m=1 s j ≥0 : M = m j=1 (1+s j )
F n,L p θ 0 ,m,s 1 ,...,sm ,

with F n,θ,m,s 1 ,...,sm the set of M -uple (k, ) of nonnegative integers with k = (k j ) j=1,...,m , = ( j,s ) j=1,...,m;s=1,...,s j such that, for all j = 1, ..., m, k j ≥ k j-1 + n θ (with convention k 0 = 0) and, for all j = 1, ..., m and all s = 1, ..., s j , 0 ≤ j,s ≤ n Lθ and, with this representation,

E   M j=1 f (Z m j )   = E   m j=1 f (Z k j ) s j s=1 f (Z k j + j,s )   . (57)
We rst study separately the following sums

(m,s)∈G M (k, )∈F n,θ,m,s 1 ,...,sm c (k, ) E   m j=1 f (Z k j ) s j s=1 f (Z k j + j,s )   ,
with G M the set of (m, s) with m ∈ {1, ..., M } and s = (s 1 , ..., s m ) with s j ≥ 0 for all j = 1, ..., m and such that M = m j=1 (s j + 1).

Let us x for the moment (m, s) ∈ G M . With the notation [START_REF] Matheron | ss trnsport in porous medi lwys di'usivec e ounterxmple[END_REF], we wish to study

(k, )∈F n,θ,m,s 1 ,...,sm E   m j=1 f (Z k j ) s j s=1 f (Z k j + j,s )   = (k, )∈F n,θ,m,s 1 ,...,sm B k, . (58)
We say that (k, ) and (k , ) belong to a same block if

∀r ∈ J , k r = k r , ∀j ∈ J , k j /d = k j /d , = .
A block is an equivalence class for this equivalence relation. We write F n,θ,m,s 1 ,...,sm for the set of (k, ) such that their block is contained in F n,θ,m,s 1 ,...,sm . We will see that the contribution of the sum over F n,θ,m,s 1 ,...,sm \ F n,θ,m,s 1 ,...,sm is neglectable in (58). Indeed, observe that if (k, ) ∈ F n,θ,m,s 1 ,...,sm \ F n,θ,m,s 1 ,...,sm , then at least one of the following condition holds true

(a) k j /d d -k j-1 < n θ ≤ ( k j /d + 1)d -1 -k j-1 if j -1 ∈ J (or k j /d d -( k j-1 /d + 1)d -d < n θ ≤ ( k j /d + 1)d -1 -k j-1 /d d if j -1 ∈ J ) (b) m ∈ J and d k m /d + max s m,s < n ≤ d( k j /d + 1) + max s m,s
Let us x J ⊂ J . Due to the rst point of Lemma 20, the contribution to (58) of blocks having a type (a) or (b) problem at indices J is in

(k j ) j ∈J , O     m j=1 n -3 4 j   J ∈{1,...,m} : #J ≥#(J \J )/2 j∈J n -1 2 +η j   = O   n LM θ n (n j ) j ∈J =n θ   m j=1 n -3 4 j   J ∈{1,...,m} : #J ≥#(J \J )/2 j∈J n -1 2 +η j   .
The study of this quantity corresponds to (59) up to replace m par m -#J and to delete indices J , which thus will be in o(n -M 8 ), as proved below. Now, using the d-block structure of F n,θ,m,s 1 ,...,sm , It follows from ( 38) that (59)

The above quantity is in

O   n LM θ J : #J ≥#(J )/2 n n 1 ,...,nm=n θ m i=1 n -3 4 i r∈J n -1 2 -η r   = O   J : #J ≥#(J )/2 n LM θ+ 1 4 (m-#(J )/2 )-( 1 4 -η)θ #(J )/2   = O n LM θ+ 1 4 (m-#J /2 )-θ 4 #J /2 +θJγ ,
where we used the fact that n r=1 r

-3 4 = O n 1 4
and that r≥n θ r -

5 4 = O n θ 4 .
Observe moreover that M = m j=1 (s j + 1) ≥ 2(m -#J ) + #J = 2m -#J , with equality if and only if s j ∈ {0, 1} for all j = 1, ..., m. It follows that

(k, )∈F n,θ,m,s 1 ,...,sm E   m j=1 f (Z k j ) s j s=1 f (Z k j + j,s )   = O n LM θ+ M 8 - M -(2m-#J ) 8 +θ #J /2 4 -#J η
In particular this is in o(n

M 8 ) as soon as M > 2m -#J or J = ∅.
This ends the proof of the rst point of Corollary 22 (since, when M is odd, we cannot have M = 2m -#J and J = ∅) and ensures that, for M even,

n -M 8 E   n k=1 f (Z k ) M   = n -M 8 (k, )∈ M p=0 F n,L p θ 0 ,M/2,1,...,1 c (k, ) E   m j=1 f (Z k j )f (Z k j + j,1 )   .
Assume from now on that θ = θ 0 and that M is even, J = ∅ and M = 2m, which means that s j = 1 for every j = 1, ..., m and let us estimate the following quantity

E n,M,θ = (k, )∈F n,θ,M/2,1,...,1 c (k, ) E   m j=1 f (Z k j )f (Z k j + j,1 )   .
Note that, when (k, ) ∈ F n,θ,M/2,1,...,1 , then c (k, ) = (2m)! 2 #{j: j =0} . Using this and applying Proposition 15 combined with the dominated convergence theorem, we obtain that

n -m 4 E n,M,θ = (2m)! 2 m n -m 4 0≤k 1 <...<km≤n:k i+1 -k i >n θ n Lθ 1 ,..., m=0 2 #{j: j >0} E   m j=1 f (Z k j )f (Z k j + j )   = (2m)! 2 m n -m 0≤k 1 <...<km≤n/d:k i+1 -k i >n θ n 3m 4 d-1 k 1 ,...,k m =0 n Lθ 1 ,..., m=0 2 #{j: j >0} E   m j=1 f (Z dk j +k j )f (Z dk j +k j + j )   + o( = (2m)! 2 m 0≤t 1 <...<tm≤1/d d m σ 2m f E det D -1 2 d 0 t 1 ,...,d 0 tm (2πσ 2 ξ ) m 2 dt 1 ...dt m + o(1) .
Therefore

lim n→+∞ n -m 4 E n,M,θ = (2m)! 2 m 0≤s 1 <...<sm≤1 σ 2m f E det D -1 2 s 1 ,...,sm (2πσ 2 ξ ) m 2 ds 1 ...ds m = (2m)!σ 2m f m!2 m (2πσ 2 ξ ) m 2 [0,1] m E det D -1 2 s 1 ,...,sm ds 1 ...ds m .
It remains now to prove that we can neglect the contribution of the (k, ) ∈ M p=1 F n,L p θ 0 ,M/2,1,...,1 \

F n,θ 0 ,M/2,1,...,1 . Fix some p = 1, ..., M . It follows from [START_REF] Marcus | wrkov proessesD qussin proessesD nd lol timesF Cambridge Studies in Advanced Mathematics[END_REF] that

n -m 4 (k, )∈F n,L p θ 0 ,M/2,1,...,1 \F n,θ 0 ,M/2,1,...,1 c (k, ) E   m j=1 f (Z k j )f (Z k j + j,1 )   = O   n -m 4 n n 1 ,...,n m-1 =n L p θ 0 m-1 i=1 n -3 4 i n θ 0 nm=1 n -3 4 m n mL p+1 θ 0   = O n -1 4 + θ 0 4 +mL p+1 θ 0 = o(1) .
The last part of Theorem 5 corresponds to the particular case f = δ 0 -δ a . In this case

σ 2 f = σ 2 0,a = k∈Z 2P(Z |k| = 0) -P(Z |k| = a) -P(Z |k| = -a) .
Appendix A. Proofs of technical lemmas for Theorem 5

Recall the context. Let M ≥ 1, θ ∈ (0, 1), η ∈ 0, 1 100 , L = κη 10M . Recall that n j = k j -k j-1 (with convention k 0 = 0). Assume n θ < n j < n and let j,1 , ..., j,s j = 0, ..., n Lθ with m j=1 (1 + s j ) = M . Proof of Lemma 16. We start by writing

p k, (a, b) = 1 (2π) M [-π,π] M e -i m j=1 [(a j -a j-1 )θ j + s j s=1 (b j,s -a j )θ j,s ] ϕ k, (θ, θ ) d(θ, θ ) . (60) But, due to the denition of d, for any u, v ∈ Z, ϕ ξ (u + 2πv d ) = (ϕ ξ ( 2π d )) v ϕ ξ (u) and so, for any u ∈ R M and v ∈ Z M , ϕ k, (u + 2π d v) = E   y∈Z ϕ ξ   m j=1 u j + 2πv j d N j (y) + s j s=1 u j,s + 2πv j,s d N j,s (y)     = E   y∈Z ϕ ξ 2π d m j=1 [v j N j (y)+ s j s=1 v j,s N j,s (y)] ϕ ξ   m j=1 u j N j (y) + s j s=1 u j,s N j,s (y)     = ϕ ξ 2π d m j=1 v j n j + s j s=1 j,s v j,s ϕ k, (u) .
and so

p k, (a, b) = 1 (2π) M [-π d , π d ] m ×[-π,π] M -m d-1 r j =0 e -i m j=1 [(a j -a j-1 )(θ j + 2πr j d )+ s j s=1 (b j,s -a j )θ j,s ϕ ξ 2π d m j=1 r j n j ϕ k, (θ, θ ) d(θ, θ ) .
Moreover, for any a ∈ Z, then Proof of Lemma 17. Due to [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Lemma 16], for any γ > 0, satises P(Ω (j) k ) = 1 -o(n -p j ) for any p > 1 and so, since n j > n θ , it follows that for all p > 1, P(Ω (j) k ) = 1 -o(n -p ). Moreover, since θ ∈ (0, θ 4 ), due to [14, Lemma 21],

∀p > 1, P det D n 1 ,...,nm < n -θ m i=1 n 3 2 i = o(n -p ) ,
uniformly on k as above.

Proof of Lemma 18. Recall that F = y ∈ Z : ∀(j, s), N j,s (y) = 0 . Due to (50), Lemma 18 follows from the following estimate

∃c > 0, {∃j,n -1 2 +η j <|θ j |} E   y∈F ϕ ξ   m j=1 θ j N j (y)   1 Ω k   dθ = o e -n c , ( 61
)
uniformly on k, as in Proposition 15. To this end, we follow and slightly adapt the proof of [START_REF] Castell | e lol limit theorem for rndom wlks in rndom senery nd on rndomly oriented lttiesF[END_REF]Proposition 10] as explained below. Observe that, up to conditioning with respect to (S k+1 -S k ) k ∈{k j-1 ,...,k j -1} , this will be a consequence of

∀j = 1, ..., m, ∀u ∈ R, n -1 2 +η j <|θ|< π d E   y∈F ϕ ξ u + θN j (y) 1 Ω k   dθ = o e -n c , (62)
uniformly on k j , j,s as above. Recall that #(Z \ F) ≤ m j=1 s j s=1 j,s ≤ M n Lθ . As in [START_REF] Castell | e lol limit theorem for rndom wlks in rndom senery nd on rndomly oriented lttiesF[END_REF]after Lemma 16], we observe that, for n large enough,

y∈F ϕ ξ (u + θN j (y)) ≤ exp - σ 2 ξ 4 n -1 2 +4γ # y : d u + θN j (y), 2π d Z ≥ n -1 4 +2γ , (63)
and that

d u + θN j (y), 2πZ d ≥ n -1 4 +2γ ⇐⇒ u θ + N j (y) ∈ I := k∈Z I k , (64)
where, for all k ∈ Z,

I k := 2kπ dθ + n -1 4 +2γ θ , 2(k + 1)π dθ - n -1 4 +2γ
θ .

In particular R \ I = k∈Z J k , where for all k ∈ Z,

J k := 2kπ dθ - n -1 4 +2γ θ , 2kπ dθ + n -1 4 +2γ
θ .

Let N ± be two positive integers such that P(X

1 = N + )P(X 1 = -N -) > 0. Let C ± = (C ± k ) k=1,...,T ∈ Z T with T = N + + N -and C + k = N + for k ≤ N -and C + k = -N -other- wise,

and symetrically and C

- k = -N -for k ≤ N + and C - k = N + otherwise.
It has been proved in [START_REF] Castell | e lol limit theorem for rndom wlks in rndom senery nd on rndomly oriented lttiesF[END_REF] (see Lemma 15 therein combined with the estimate P(D n ) = 1 -o(e -cn ) in Section 2.8 therein) that, for n large enough,

P(Ω k \ E j ) = o(e -cn j ) ,
(65) with

E j = #{y ∈ Z : C j (y) ≥ n 1 2 -2γ j } ≥ 3N + N -n 1 2 -2γ j ,
and where, for any y ∈ Z,

C j (y) := # k = 0, . . . , n j T -1 : S k j-1 +kT -S k j-1 = y and (X k j-1 +kT , . . . , X k j-1 +(k+1)T -1 ) = C ± . Now, on E j , we dene Y i for i = 1, . . . , n 1 2 -2γ j
, by

Y 1 := min y ∈ Z : C j (y) ≥ n 1 2 -2γ j ,
and

Y i+1 := min y ≥ Y i + 3N -N + : C j (y) ≥ n 1 2 -2γ j for i ≥ 1.
For every i = 1, . . . , n

1 2 -2γ j , let t 1 i , . . . , t n 1 2 -2γ j i be the n 1 2 -2γ j
rst times (which are multiples of T ) when a peak of the form C ± is based on the site Y i . We also dene N 0 j (Y i + N + N -) as the number of visits of (S k j-1 +k -S k j-1 ) k≥0 before time n j to Y i + N + N -, which do not occur during the time intervals

[t u i , t u i + T ], for u ≤ n 1 2 -2γ j
. We proved in [START_REF] Castell | e lol limit theorem for rndom wlks in rndom senery nd on rndomly oriented lttiesF[END_REF]Lemma 16] that, for any H ≥ 0,

P u θ + N j (Y i + N + N -) ∈ I E n , N 0 j (Y i + N + N -) = H = P H + u θ + b j ∈ I , (66)
where b j is a random variable with binomial distribution B n Thus, conditionally to (S k+1 -S k ) k ∈{k j-1 ,...,k j -1} , E j and ((N 0 j (Y i + N + N -), i ≥ 1), the events

{ u θ + N j (Y i + N + N -) ∈ I}, i ≥ 1,
are independent of each other, and all happen with probability at least 1/3. We conclude that

P   E j ∩    # i : u θ + N j (Y i + N + N -) ∈ I ≤ n 1 2 -2γ j 4      ≤ P   B j ≤ n 1 2 -2γ j 4   = o(e -c n j ) , (67)
where

B j has binomial distribution B n 1 2 -2γ j ; 1 3 . But if #{y ∈ Z : N j (z) ∈ I} ≥ n 1 2 -2γ j /4
, then, by ( 63) and ( 64) there exists a constant c > 0, such that, for any n large enough,

y∈F |ϕ ξ (u + θN j (y))| ≤ exp -c n 1 2 -2γ j n -1 2 +4γ j , since #(Z \ F) n 1 2 -2γ j /4
. This, combined with (65) and (67), ends the proof of (62) and so of Lemma 18.

Proof of Lemma 19. We have to estimate B k, ,I

k ,Ω k uniformly on k, as in Proposition 15, where

I (2) k = V k × [-π, π] M -m
and where V k is the set of θ ∈ R m such that for all j = 1, ..., m,

|θ j | < n -1 2 +η j
and such that there exists some j 0 = 1, ..., m satisfying n

-1 2 -η j 0 < |θ j 0 |. Let ε 0 > 0 be such that ∀u ∈ [-ε 0 , ε 0 ], |ϕ ξ (u)| ≤ e - σ 2 ξ u 2 4
.

(68) We dene the events

H k = Ω k ∩ {∀y ∈ Z, | m j=1 θ j N j (y)| ≤ ε 0 /2} and H k :=    #    y ∈ Z : m j=1 θ j N j (y) ∈ ε 0 4 , ε 0 2    > n 1 4    .
Due to [14, Lemma 21 and last formula of p. 2446],

∃c > 0, P Ω k \ (H k ∪ H k ) = O   m j=1 n -3 4 j   ,
uniformly on k as above and uniformly on θ ∈ V k . Thus,

B k, ,I (2) k ,Ω k \(H k ∪H k ) = O   m j=1 n -5 4 +η j   , (69)
where we used the fact that V k dθ ≤ m j=1 n -1 2 +η j . Moreover, for n large enough, it follows from the denition of H k , from ( 51) and (68) that

B k, ,I (2) k ,Ω k ∩H k ) = O   V k E   y∈F ϕ ξ   m j=1 θ j N j (y)   1 Ω k ∩H k   dθ   ≤ e - σ 2 ξ ε 2 0 n 1 4 64 . (70)
Finally, it remains to estimate B k, ,I

k ,Ω k ∩H k

. To this end we write

V k E   y∈F ϕ ξ   m j=1 θ j N j (y)   1 Ω k ∩H k   dθ ≤ V k E e - σ 2 ξ 4 y∈F ( m j=1 θ j N j (y)) 2 1 Ω k dθ ≤ V k   m j=1 n -3 4 j   E   e - σ 2 ξ 4 y∈F m j=1 θ j n -3 4 j N j (y) 2 1 Ω k   dθ ≤   m j=1 n -3 4 j   E ( D k ) 1 2 V k (det D k ) -1 2 e - σ 2 ξ |v| 2 4 1 Ω k dv , (71)
with the successive changes of variable

θ j = n 3 4 j θ j and v = ( D k ) 1 2 θ , with D k =   (n i n j ) -3 4 y∈F N i (y)N j (y)   i,j
and V k = Diag(n

3 4 i )V k .
Note that V k is the set of (θ 1 , ..., θ m ) such that |θ j | ≤ n

k := (n i n j ) -3 4 y∈Z N i (y)N j (y) i,j
. To this end, writing Σ m for the set of permutations of the set {1, ..., m} and κ(σ) for the signature of σ ∈ Σ m , we observe that, on Ω k ,

det D k -det D k =   m j=1 n -3 2 j   σ∈Σm (-1) κ(σ) m j=1   y∈F N j (y)N σ(j) (y)   - σ∈Σm (-1) κ(σ) m j=1   y∈Z N j (y)N σ(j) (y)   ≤   m j=1 n -3 2 j   σ∈Σm m j=1 z∈Z\F N j (z)N σ(j) (z) j =j   y∈Z N j (y)N σ(j ) (y)   ≤   m j=1 n -3 2 j   σ∈Σm m j=1 #(Z \ F)n 1+2γ 2 j n 1+2γ 2 σ(j) j =j V j V σ(j ) ,
where we used the Cauchy-Schwarz inequality together with the notations and estimates given after Lemma 17. Using ( 45) and ( 51), it follows that, on Ω k ,

det D k -det D k   m j=1 n -3 2 j   n Lθ m j=1 n 1+2γ 2 j n 1+2γ 2 σ(j) j =j n 3(1+2γ) 4 j n 3(1+2γ) 4 σ(j ) 1 2 n mγ-θ 2 +Lθ n -θ -(M +1)Lθ ≤ n -(M +1)Lθ 2 det D k , since θ ≤ θ 2 -2M Lθ < θ 2 -mγ-M
Lθ and where we used the fact that det

D k = det D k m j=1 n -3 2 j
together with the denition of Ω k . Therefore, on Ω k , det D k ≥ 1 2 det D k . Thus, due to (71),

V k E   y∈F ϕ ξ   m j=1 θ j N j (y)   1 Ω k ∩H k   dθ ≤ O     m j=1 n -3 4 j   E ( D k ) 1 2 V k (det D k ) -1 2 1 Ω k e - σ 2 ξ |v| 2 4 dv   = O     m j=1 n -3 4 j   E (det D k ) -1 2 1 Ω k ( D k ) 1 2 V k e - σ 2 ξ |v| 2 4 dv   , (72)
By denition of V k , for any v ∈ ( D k ) 1 2 V k , |v| 2 ≥ ( λ k ) 1 2 n ( 1 4 -η)θ
, where λ k is the smallest eigenvalue of D k . Since all the eigenvalues of D k are nonnegative ( D k being symmetric and nonnegative), it follows that all the eigenvalues of D k are smaller than trace

( D k ) ≤ m j=1 V j n 3 2 j ≤ mn 3γ (on Ω k ). Thus, on Ω k , ( λ k ) 1 2 n ( 1 4 -η)θ ≥ det( D k ) 1 2
(m

1 2 n 3γ 2 ) m-1 n ( 1 4 -η)θ ≥ n ( 1 4 -η)θ-θ 2 - 3γ(m-1) 2 2m m-1 2 n θ 16 , (73)
since ηθ, θ 2 , and 3γ(m-1) 2 are all strictly smaller θ 16 . Hence

E (det D k ) -1 2 1 Ω k ( D k ) 1 2 V k e - σ 2 ξ |v| 2 4 dv ≤ E (det D k ) -1 2 1 Ω k |v| 2 >n θ 16 e - σ 2 ξ |v| 2 4 dv = O n -p ,
for any p > 0. This combined with (69), ( 70) and (72) ends the proof of the lemma. It will be worthwhile to note that the previous estimate also holds true when λ k is replaced by the smallest eigenvalue

λ k of D k .
Before proving Lemma 20, we state a useful coupling lemma allowing us to replace det D k by a copy independent of (N j,s ) j,s . Up to enlarging the probability space if necessary, we consider X = (X k ) k≥1 an independent copy of the increments X = (X k ) k≥0 of the random walk S. We then dene the random walk S as follows:

S m = m k=1 X k with X k = X k if k j-1 + j-1 ≤ k < k j and X k = X k if k j ≤ k < k j + j
, with j := max s=1,...,s j j,s . We dene Ω k , N j and D k for the space as we have dened Ω k , N j , D k (up to replace S by S ). Lemma 23. There exists

Ω k ⊂ Ω k ∩ Ω k such that ∀p > 0, P (Ω k ∩ Ω k ) \ Ω k = O(n -p ) (74) and such that, on Ω k , (det D k ) -1 2 -(det D k ) -1 2 ≤ n -θ 8 -Lθ (det D -3 2 k + (det D k ) -3 2 ) . Moreover E ((det D k ) -1 2 -(det D k ) -1 2 )1 Ω k ≤ n -θ 8 -Lθ m j=1 n -9 4 j . (75)
Proof of Lemma 23. Observe that

S k j -S k j = h j = j <j S k j + j -S k j -(S k j + j -S k j )
and, on

Ω k ∩ Ω k , |N j (z) -N j (z)| = |N j (z) -N j (z + h j )| ≤ n 1 4 + γ 2 j |h j | 1 2 ,
for all z ∈ Z \ k j-1 + j m=k j-1 {S m , S m }.

We will prove that det D k is close enough to det D k = det( y∈Z N i (y)N j (y)). Due to the Markov inequality,

∀p > 0, P |S j | > h ≤ O   p 2 j h p   = O n -γ p ,
where we set h = n γ + κθη 20M ≥ n γ Moreover

V j := y∈Z (N j (y)) 2 ≤ y∈Z (N j (y)) 2 + 2 3 j ≤ n 3 2 +3γ j .
This allows us to observe that, on Ω k ,

det D k -det D k = σ∈Σm (-1) κ(σ) m j=1   y∈Z N j (y)(N σ(j) (y)   - σ∈Σm (-1) κ(σ) m j=1   y∈Z N j (y)N σ(j) (y)   ≤ σ∈Σm m j=1 z∈Z [N j (z)N σ(j) (z) -N j (z)N σ(j) (z)] j =j max   y∈Z N j (y)N σ(j ) (y), y∈Z N j (y)N σ(j ) (y)   ≤ 3n γ 2 + κθη 40M σ∈Σm m j=1 V 1 2 j n 1 2 +γ σ(j) + (V σ(j) ) 1 2 n 1 2 +γ j j =j max V j V σ(j ) , V j V σ(j ) 1 2 ≤ 3n γ 2 + κθη 40M m! m j =1 n 3 2 +3γ j m j=1 n -1 4 -γ 2 j m j =1 n 3 2 j m j=1 n -1 8 j n -Lθ , since Lθ + 3mγ -θ 4 + γ 2 < -θ
8 -Lθ, and so, on Ω k ,

(det D k ) -1 2 -(det D k ) -1 2 ≤ n -θ 8 -Lθ (det D -3 2 k + (det D k ) -3 2 ) .
We conclude thanks to [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Lemma 21] which ensures that

E (det D k ) -3 2 1 Ω k = O m j=1 n -9 4 j .
The proof of Lemma 20 will also use the following result. Recall that we set J = {j = 1, ..., m : s j = 0} and that J ⊂ J . Lemma 24. Under the assumptions of Lemma 20,

k j =0,...,d-1, ∀j∈J B k+k , ,I (3) k ,Ω k = d m (2π) M I (3) k E 1 Ω k F (θ, θ )G(θ, θ ) d(θ, θ ) ,
with k ∈ Z m such that k j = 0 for all j ∈ J , and with

G(θ, θ ) := j ∈J   a j ∈αk j +dZ b j,s ,...,b j,s j ∈Z f (a j ) s j v=1 f (b j,v ) e ia j (θ j+1 -θ j )-i s j s=1 (b j,s -a j )θ j,s   × y∈Z\S ϕ ξ   m j=1 θ j N j (y) + s j s=1 θ j,s N j,s (y))   ,
with S = j∈J S j , S j := {S k j , ..., S k j +d-1 }, so that {S k j , ..., S k j +d-1 } and, uniformly on k,

and on Ω k , F (θ, θ ) = O J ⊂J j∈J \J (|θ j | + |θ j+1 |)1 j∈J B j ) with B j = 1 S j ∩ j ∈J \{j} S j =∅ . If a∈Z f (b + dZ) = 0 for all b ∈ Z (true if d = 1), then F (θ, θ ) = O j∈J (|θ j | + |θ j+1 |)
(with convention θ m+1 = 0).

Proof. We start by writing

k j =0,...,d-1, ∀j∈J B k+k , ,I (3) k ,Ω k = d m (2π) M I (3) k E 1 Ω k F (θ, θ )G(θ, θ ) d(θ, θ ) , (76)
where we set

F (θ, θ ) := k j =0,...,d-1, ∀j∈J j∈J   a j ∈(k j +k j )α+dZ f (a j )e -ia j (θ j -θ j+1 )   × y∈S ϕ ξ m r=1 θ r N r,k (y) + sr s=1 θ r,s N r,s (y)) , with N r,k (y) = #{u = k r-1 + k r-1 , ..., k r + k r -1 : S u = y} .
If we had a∈u+dZ f (a) = 0 for all u ∈ Z, the proof of Lemma 24 will be ended by noticing that a j ∈(k j +k j )α+dZ f (a j )e -ia j (θ j -θ j+1 ) = a j ∈(k j +k j )α+dZ f (a j ) e -ia j (θ j -θ j+1 ) -1 ,

which is in O (|θ j | + |θ j+1 |) since a∈Z |af (a)| < ∞.
Since we just assume here that a∈Z f (a) = 0, we need a more delicate approach. We rewrite F as follows except maybe if r ∈ J and y ∈ S r or if r -1 ∈ J and y ∈ S r-1 . We order the elements of J as follows: j 1 < ... < j J and write

F (θ, θ ) := k j =0,...,d-1, ∀j∈J a j ∈(k j +k j )α+dZ ∀j∈J   j∈J H j,k j (θ j -θ j+1 )   Ψ(k ) with H j,k j (θ) := a j ∈(k j +k j )α+dZ f (a j )e -ia j θ , Ψ (k )
F (θ, θ ) = F 0 (θ, θ ) + F 1 (θ, θ )
with

F 1 (θ, θ ) = d-1 k j 1 =0 H j 1 ,k j 1 (0) d-1 k j 2 ,...,k j J =0   j∈J \{1} H j,k j (θ j -θ j+1 )   Ψ(k )
and

F 0 (θ, θ ) = d-1 k j 1 =0 H j 1 ,k j 1 (θ j 1 -θ j 1 +1 ) -H j 1 ,k j 1 (0) d-1 k j 2 ,...,k j J =0   j∈J \{1} H j,k j (θ j -θ j+1 )   Ψ(k ) .
Note that H j 1 ,k j 1

(θ j 1 -θ j 1 +1 ) -H j 1 ,k j 1 (0) is in O(|θ j | + |θ j+1 |).
Since a f (a) = 0, F 1 satises

F 1 (θ, θ ) = d-1 k j 1 =0 H j 1 ,k j 1 (0) d-1 k j 2 ,...,k j J =0   j∈J \{j 1 } H j,k j (θ j -θ j+1 )   ∆ j 1 Ψ(k ) with ∆ j φ(k ) = φ(k ) -φ(k j ),
where k j ∈ N m is such that (k j ) i = k i for i = j, and (k j ) j = 0.

Proceding iteratively on J , we obtain

F (θ, θ ) = 1 ,..., J ∈{0,1} F 1 ,..., J (θ, θ ) , (77)
with

F 1 ,..., J (θ, θ ) =   j : j =0 H j 1 ,k j 1 (θ j 1 -θ j 1 +1 ) -H j 1 ,k j 1 (0)     j: j =1 H j,k j (0)   ∆ J j J • • • ∆ 1 j 1 Ψ(k ) ,
with convention ∆ 0 j = Id. The rst part will be easily dominated by O j : j =0 (|θ j | + |θ j +1 |) . Let us study the second part of the formula exploiting the fact that a∈Z f (a) = 0. The diculty here is that k appears both in j: j =1 H j,k j (0) and in ∆ ... ψ(k ). The value of ( 1 , ..., J ) being xed, we consider the set J of the j ∈ J such that j = 1. Observe that, if S j ∩ S j = ∅, then

∆ j ∆ j Ψ(k ) = ∆ j Ψ S \S j (k j ) ∆ j Ψ S j ( k j ) with Ψ S 0 (k ) = y∈S 0 ϕ ξ m r=1 θ r N r,k (y) + sr s=1 θ r,s N r,s (y)) ,
and where we set k j for the vector of Z m with j-th coordinate equal to k j , all the other coordinates being null. Let J 0 be the set of j ∈ J such that S j ∩ j ∈S \{j} S j = ∅. Then

k j =0,...,d-1, ∀j∈J 0   j∈J 0 H j,k j (0)   ∆ J j J • • • ∆ 1 j 1 Ψ(k ) = j∈J 0   d-1 k j =0 H j,k j (0)∆ j Ψ S j k j   ∆ J \J 0 Ψ(k S 0 )
with k S 0 ∈ N m such that (k j ) i = k i for i ∈ S 0 , the other coordinates being null, the notation ∆ J \J 0 standing for the composition of all the operators ∆ j for j ∈ J \ J 0 . We conclude by using (77) and by noticing that

  j ∈S\S H j 1 ,k j 1 (θ j 1 -θ j 1 +1 ) -H j 1 ,k j 1 (0)   = O   j ∈S\S (|θ j | + |θ j +1 |)   , j∈J 0   d-1 k j =0 H j,k j (0)∆ j Ψ S j k j   = O   j ∈S\S (|θ j | + |θ j +1 |)   and that j ∈ J \ J 0 =⇒ S j ∩ j ∈S 0 \{j} S j .
The following lemma will be useful to estimate the term F appearing in Lemma 24. It is not needed when a∈Z f (b + ad) = 0 for all b ∈ Z.

Lemma 25. For any J ⊂ J ,

P   Ω k ∩ j∈J B j   = O   J ⊂J \{min J }, J ≥#J /2 n Jγ j∈J (k j -k - j ) -1 2   ,
where k -

j = max{k s ≤ k j , s ∈ J }.
Proof. It is enough to study

P   Ω k ∩ j∈J S k j +r j = S k m(j) +s j  
for any m(j) ∈ J \ {j}, r j , s j ∈ {0, ..., d -1}. This probability is dominated by

P Ω k ∩ ∀j ∈ J , |S k j -S k m(j) | ≤ n v + o(n -p ) ,
for all p, v > 0. We partition the set J by the equivalence relation generated by the relation j ∼ m j . We write R(j) for the class of j and R for the set of these equivalence classes. Observe that the number of equivalent classes is at most #J /2 . We order the set J in j 1 < ... < j J . We wish to estimate Ar, r∈R

P Ω k , ∀i = 1, ..., J -1, S k j i+1 -k j i = A R(j i+1 ) -A R(j i ) + O(n v ) ,
where the sum is over

(A r ) r∈R ∈ Z R such that A R(1) = 0, A R(j i+1 ) -A R(j i ) = O((k j i+1 -k j i ) 1 2 + γ 2 ).
Due to the local limit theorem and the independence of the increments of S, the above probability is in

Ar, r∈R J -1 i=1 n v O (k j i+1 -k j i ) -1 2 .
Now let us control the cardinal of the admissible (A r , r ∈ R). To this end, consider the set J of the smallest representants of R. Then the above quantity is smaller than

n J (v+ γ 2 ) j∈J \J (k j -k - j ) -1 2 .
Proof of Lemma 20. All the estimates below are uniformly in k. For the rst estimate, we have to estimate the following integral

∀j,|θ j |<n -1 2 -η j   j ∈J   a j ∈αk j +dZ f (a j )e ia j (θ j+1 -θ j ) s j s=1 b j,s ∈Z f (b j,s )e -i(b j,s -a j )θ j,s     × E   1 Ω k F (θ, θ ) y∈Z\S A y   dθ , (78)
where we set

Let us study

E k, (θ, θ ) := y∈Z\S A y - y∈Z\S B y , (79) with
B y := exp   - σ 2 ξ 2   m j=1 θ j N j (y)   2   ϕ ξ   m j=1 s j s=1 θ j,s N j,s (y)   . But, on Ω k , if |θ j | ≤ n - 1
2 -η j for all j = 1, ..., m, and so

∀y ∈ Z, m j=1 θ j N j (y) ≤ m j=1 |θ j |N * j ≤ m j=1 n -η 2 j ≤ mn -θη 2 < ε 0 , (80)
as soon as n is large enough (uniformly on

n j ∈ [n θ , n]). Thus |E k, (θ, θ )| is dominated by y∈Z |A y -B y | e - σ 2 ξ 4 z∈F \(S ∪{y}) ( m j=1 θ j N j (z))
for n large enough. Now, on Ω k , according to [START_REF] Zweimüller | wixing limit theorems for ergodi trnsformtions[END_REF],

∀y ∈ Z, z∈F \(S ∪{y})   m j=1 θ j N j (z)   2 ≥ z ∈Z   m j=1 θ j N j (z )   2 -M (d + n ηθ 10M )n -θη . (81) It follows that |E k, (θ, θ )| ≤ (A + B) exp   - σ 2 ξ 4 z ∈Z   m j=1 θ j N j (z )   2 -O n -9θ 10η   , (82)
with

A := y∈F \S ϕ ξ   m j=1 θ j N j (y)   -e - σ 2 ξ 2 ( m j=1 θ j N j (y)) 2 ≤ y∈Z m j=1 θ j N j (y) 2 C n -κθη 2 ( 83
)
where we used the fact that

ϕ ξ (u) -exp - σ 2 ξ |u| 2 2 ≤ |u| 2+κ for all u ∈ R,
since ξ admits a moment of order 2 + κ and there exists C 0 > 0 such that

B := y∈Z\F ϕ ξ   m j=1 θ j N j (y) + s j s=1 θ j,s N j,s (y)   -e - σ 2 ξ 2 ( m j=1 θ j N j (y)) 2 ϕ ξ   m j=1 s j s=1 θ j,s N j,s (y)   ≤ C 0 y∈Z\F m j=1 θ j N j (y) ≤ C 0 m j=1 s j s=1 j,s n -θη 2 = O n θη 10M -θη 2 = O n -θη 4 , (84)
since ϕ ξ and u → e -u 2 2 are Lipschitz continuous. Recall that it has been proved in [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Lemma 21] that

E | det D k | -1 2 1 Ω k = O   m j=1 n -3 4 j   , (85)
uniformly on k.

Combining Lemmas 24 and ( 25), (82), (83), (84), (85) and using the change of variable

v = (D k ) 1 2 θ with D k = y∈Z N i (y)N j (y) i,j
, it follows that there exists

C 1 > 0 such that ∀j,|θ j |≤n -1 2 -η j E F (θ, θ )E k, (θ, θ ) 1 Ω k d(θ, θ ) ≤ C 1 R m n -κθη 2 |v| 2 2 + O(n -θη 4 ) e - σ 2 ξ |v| 2 4 dv J 0 ⊂J j∈J \J 0 n -1 2 -η j + n -1 2 -η j+1 E | det D k | -1 2 1 Ω k ∩ j∈J 0 B j = O   n -κθη 4   m j=1 n -3 4 j   E k (J )   , (86)
with

E k (J ) = J ⊂J j∈J \J n -1 2 -η j + n -1 2 -η j+1 J 0 ⊂J \{min J }, J 0 ≥#J /2 n Jγ+ θ 2   j∈J 0 (k j -k - j ) -1 2   = O   J ⊂J ∪(J +1) : #J ≥#J /2   j∈J n -1 2 +η j     .
where k - j = max{k s ≤ k j , s ∈ J }. Combining this last estimate with (78) and Lemmas 24 and 25,

k j =0,...,d-1, ∀j∈J B k+k , ,I (3) k ,Ω k = d m (2π) M (a j ) j ∈J ,(b j,s ) j,s 1 {∀i ∈J , a i ∈k i α+dZ} [-π,π] M -m E [I 1 (a) I 2 (a, b)1 Ω k ] dθ + O   n -κθη 4 m j=1 n -3 4 j E k (J )   , (87)
with

I 1 (a) := ∀j, |θ j |≤n -1 2 -η j   j ∈J e -i m j= (a j -a j-1 )θ j   F (θ, θ )e - σ 2 ξ 2 y∈Z\S ( m j=1 θ j N j (y)) 2 dθ = O ∀j, |θ j |≤n -1 2 -η j F (θ, θ )e - σ 2 ξ 2 ( y∈Z ( m j=1 θ j N j (y)) 2 -M dn -η j ) dθ = O det D -1 2 k sup θ∈V k F (θ, θ ) R m e - σ 2 ξ |v| 2 2 2 dv , (88) with the change of variable v = D 1 2 k θ and I 2 (a, b) :=   j ∈J f (a j ) s j s=1 f (b j,s )e -i j,s (b j,s -a j )θ j,s   y∈Z\S ϕ ξ   j,s (θ j,s N j,s (y))   = O   j ∈J f (a j ) s j s=1 f (b j,s )   . ( 89
)
Since a∈Z |f (a)| < ∞, it follows from(85), (87), ( 88) and (89) that

k j =0,...,d-1, ∀j∈J B k+k , ,I (3) k ,Ω k = O   E k (J )   m j=1 n -3 4 j     .
This ends the proof of the rst point of Lemma 20. Assume now that s j = 1 for all j = 1, ..., m (in particular J = ∅). Then

I 1 (a) = ∀j, |θ j |≤n -1 2 -η j e -i m j=1 (a j -a j-1 )θ j e - σ 2 ξ 2 y∈Z\S ( m j=1 θ j N j (y)) 2 dθ =   m j=1 n -3 4 j   ∀j,|θ j |≤n 1 4 -η j e -i m j=1 n -3 4 j (a j -a j-1 )θ j e - σ 2 ξ 2 y∈Z ( m j=1 θ j n -3 4 j N j (y)) 2 dθ =   m j=1 n -3 4 j   D 1 2 k U k (det D k ) -1 2 e -i D -1 2 k (n -3 4 j (a j -a j-1 )) j ,v e - σ 2 ξ |v| 2 2 2 dv ,
where U k is the set of θ = (θ 1 , ..., θ m ) such that |θ j | ≤ n 1 4 -η j for all j = 1, ..., m and with

D k = (n i n j ) -3 4 y∈Z N i (y)N j (y) i,j . Moreover I 2 (a, b) = (2π) m j=1 s j   m j=1 (f (a j )f (b j,1 ))   P   ∀j, y∈Z\S N j,1 (y)ξ y = b j,1 -a j (N j,1 ) j   = (2π) M -m   m j=1 f (a j )   E   f   a j + y∈Z N j,1 (y)ξ y   1 {a j + y∈Z N j,1 (y)ξy=b j,1 } (N j,1 ) j   .
Thus, it follows that, uniformly in k and on Ω k ,

d m (2π) M b 1,1 ,...,b m,1 ∈Z I 1 (a) I 2 (a, b) = d 2π m   m j=1 f (a j )   (det D k ) -1 2 R m e -i D -1 2 k (n -3 4 j (a j -a j-1 )) j ,v e - σ 2 ξ |v| 2 2 2 dv + O(n -p ) E   f   a j + y∈Z N j,1 (y)ξ y   (N j,1 ) j  
for all p > 0, as seen at the end of the proof of Lemma 19 (applied with D k ) and so

j n j ) -3 2 λ -1 k 1 Ω k , (92)
where λ k is the smallest eigenvalue of D k . For the last term, we use (73) (applied for D k ), which ensures that on Ω k ,

λ k ≥ det D k (mn 3γ ) m-1
and so (min

j n j ) -3 2 E (det D k ) -1 2 λ -1 k 1 Ω k ≤ (mn 3γ
where we used [14, Lemma 21] which ensures that E (det D k ) - The last step of the proof of the lemma consists in studying the following quantity and that Let f : Z → R be such that a∈Z |f (a)| < ∞. In this appendix we prove that all the moments of n -1 4 n-1 k=0 f (Z k ) converge to those of a∈Z f (a)σ -1 ξ L 1 (0), as n → +∞.

G k :=E   (det D k ) -1 2 1 Ω k m j=1 f (a j )f   a j + y∈Z N j,s (
E (det D k ) -1 2 1 Ω k ∼ n -3m 4 E det D - 1
Due to Theorem 1, it is enough to prove the convergence of every moment. The key result is the following proposition. as n → +∞ and n i /n → T i , where D t 1 ,...,t k = ( R L t i (x)L t j (x) dx) i,j=1,...,k where L is the local time of the brownian motion B, limit of (S nt / √ n) t as n goes to innity. Moreover, for every k ≥ 1 and every ϑ ∈ (0, 1), there exists C = C(k, θ) > 0, such that Proof. The lemma has been proved for a i ≡ 0 in [14, Theorem 5]. The proof in the general case is the straighforward adaptation of [14, Section 5]. For completness, we explain the required adaptations. The proof of the present result follows line by line the same proof with the adjonction of a term e -i k j=1 (a j -a j-1 )θ j (with convention a 0 = 0) in the integrals appearing in [14, Lemma 15] (see Lemma 16 with M = m = k and s j ≡ 0). Lemma 16 (denition of the good set) and Propositions 18 and 19 (estimates of the integral of the absolute values) of [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF] are unchanged. The only dierence in the proof concern [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Proposition 17] and more specically [14, Lemma 23] for which the there is a multiplication by e -i k j=1 (a j -a j-1 )θ j in the integral. The only dierence in the proof of [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Lemma 23] is that the quantity I n 1 ,...,n k considered therein (n i corresponding to nT i -nT i-1 ) is slightly modied with the multiplication in the integral by a quantity converging in probability to 1 (with the notations of the proof of [START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF]Lemma 23]. Indeed, considering the real part of the integral, this quantity is cos( k j=0 (a j -a j-1 )(A -1 2 n 1 ,...,n k r) j ) (with the notations of [14, Lemma 23]) which is equal to 1 up to an error in O min 1, µ -1 n 1 ,...,n k |r| 2 where µ n 1 ,...,n k is the smallest eigenvalue of A n 1 ,...,n k , which is proved to converges to 0 in [14, Lemma 23], and so the asymptotic behaviour of I n 1 ,...,n k is the same as when a j ≡ 0.

Proof of the convergence of moments in Theorem 3. Take ϑ < 1 4 . Note that the last point of the lemma ensures that

P [Z n 1 = a 1 , . . . Z n 1 +•••+n k = a k ] ≤ C   i:n i >n ϑ n i   -3/4
. Let α 0 be such that αα 0 ∈ 1 + dZ. Then a i = q i α + dZ is equivalent to q i ∈ a i α 0 + dZ. Thus

E      n-1 q=0 f (Z q )   k    = n-1 q 1 ,...,q k =0
E [f (Z q 1 )...f (Z q k )] = a 1 ,...,a k ∈Z f (a 1 )...f (a k ) n-1 q 1 ,...,q k =0 P(Z q 1 = a 1 , ..., Z q k = a

Z

  n := a∈Z f (a)N n (a) . 1 Indeed d ≥ 1 is such that {u : |ϕ ξ (u)| = 1} = (2π/d)Z and a.s. e 2iπξ d = e 2iπα d which is a primitive d-th root of the unity.

  setting a i := a i / √ u, and making the change of variable y = x/ √ u, with h i (x) = g i ( √ ux) and so (13) becomes

n≥0 4 n- 1 k=0

 41 converges almost surely to I( f ), and we have proved the rst part of the proposition. The second part comes from the rst part combined with (5) and the Slustky theorem.Proof of Theorem 4. Proposition 12 states that n -1 f • T k converges in distribution, with respect to µ ⊗ δ 0 µ, to Ω f d µ σ -1 ξ L 1 (0). Thus, Theorem 4 follows from Proposition 12 combined with [51,Theorem 1].

Lemma 21 .

 21 Under the assumptions 2 of Theorem 5,≥1 d-1 =0 a,b∈Z f (a)f (b)P(Z + d = b -a) < ∞

  since a,b∈Z f (a)f (b) = 0, a∈Z |af (a)| < ∞ and using the change of variable v = tV1 2

  (k, )∈F n,θ,m,s 1 ,...,sm B (k, ) ...,m} : #J ≥(#J )/2 j∈J n

- 1 r=0 e -2iaπr d ϕ ξ 2π d vr = 0 except if e -2iaπ d ϕ ξ 2π d v = 1

 11 (i.e. if vα -a ∈ dZ) and then this sum is equal to d. This ends the proof of Lemma 16.

13 ,

 13 Lemmas 17 and 18] (see in particular the last formula in the proof of Lemma 17) that∀H ∈ R, P H + b n ∈ I ≥ 1 3 .

  there exists j 0 = 1, ..., m such that|θ j 0 | ≥ n 1 4 -η j 0 .Let us prove that, in the above formula, we can approximate the determinant of D k by the one of D

1 2 j

 12 . Thus we setΩ k := Ω k ∩ Ω k ∩ {∀j = 1, ..., m, |h j | ≤ h}and we observe that P ((Ω k ∩ Ω k ) \ Ω k ) = O(n -p ) for all p > 0. Moreover, on Ω k , |N j (z) -N j (z)| ≤ 2 j + n

  s N r,s (y)) ,and with N r,k (y) = #{u = k r-1 + k r-1 , ..., k r + k r -1 : S u = y}. Note that N r,k (y) = N r (y)

j-κθη 4 +

 4 1 ,...,b m,1 ∈Z I 1 (a) I 2 (a, b) (a j -a j-1 )) j , v j + y∈Z N j,s (y)ξ y   (N j,1 )   ,for all p. Due to (87), we obtain thatB k, ,I (3) E (det D k ) -1 2 (min

3 2 1 1 4 and

 114 Ω k = O (1) uniformly in k.This combined with (92) implies that B k, ,I (3) ,...,am∈Z1 {∀i, a i =k i α+dZ} E   (det D k ) since L(M + 1)θ < 3θ 2 -3(m -1)γ.

= 1 Moreover [ 14 ,

 114 E (det D k ) -1 2 1 Ω k m j=1 f (a j )E   f   a j + y∈Z N j,s(y)ξ y the fact that D k has the same distribution as D k and is independent of N j,s . This combined with (95), (93), (74) and (75) ensures that B k, ,I (3) ,...,am∈Z1 {∀i, a i =k i α+dZ} E (det D k ) -1 2 1 Ω k m j=1f (a j )E f a j + Z j + O Lemmas 21and 23] ensure thatE (det D k ) -1 2 1 Ω k = O

Proposition 26 . 2 T 1

 2621 For all a 1 , ..., a k ∈ Z,P (Z n 1 = a 1 , . . . , Z n k = a k ) ∼ 1 {∀i, a i ∈n i α+dZ} ,...,T k ] n -3k/4 ,

P,

  [Z n 1 = a 1 , . . . , Z n 1 +•••+n k = a k ] ≤ C for all n ≥ 1, all a 1 , ..., a k ∈ Z and all n 1 , . . . , n k ∈ [n ϑ , n].

r 1 - 1 q 1 a 1 - 1 q 1 a 1 4 a 1 t 1 t 1

 11111114111 ,...,r k =0 a 1 ,...a k ∈Z f (a 1 )...f (a k ) n d ,...,q k =0 P(Z r 1 +q 1 d = a 1 , ..., Z r k +q k d = a k ) ,...a k ∈Z f (a 1 )...f (a k ) n d ,...,q k =0 P(Z a 1 α 0 +q 1 d = a 1 , ..., Z a k α 0 +q k d = a k ) , with x the representant of x + dZ belonging to {0, ..., d -1}. It follows that ,...,a k ∈Z f (a 1 )...f (a k )n k H k = o(n k 4 ) + a 1 ,...,a k ∈Z f (a 1 )...f (a k )n k H k , with H k := [0,1/d] k P Z a 1 α 0 + t 1 n d = a 1 , ..., Z a k α 0 + t k n d = a k dt 1 ...dt k H k = [0,1/d] k n 3k 4 P Z a 1 α 0 + t 1 n d = a 1 , ..., Z a k α 0 + t k n d = a k 1 min i,j | t i n -t j n |>2n ϑ dt 1 ...dt k .Due to the dominated convergence theorem, we conclude that ,...,a k ∈Z f (a 1 )...f (a k ) d,...,t k d ]dt 1 ...dt k d,...,t k d ]dt 1 ...dt k L 1 (0)) k ] ,due to[START_REF] Castell | yn the lol time of rndom proesses in rndom seneryF[END_REF] Theorem 3].

  D t 1 ,...,tm := R L t i (x)L t j (x) dx i,j=1,...,m where (L t (x)) t≥0,x∈R is the local time of the Brownian motion B. Since det D t 1 ,...,tm is a Gram determinant, we have the iterative relation

	1	1
	det D t 1 ,...,t m+1 = det D 2 t 1 ,...,tm d(L t m+1 , V ect(L t 1 , ..., L tm )) , 2

  2 t 1 ,...,tm as k j /n → t j and n → +∞. This ends the proof of the lemma.

Appendix B. Moment convergence in Theorem 3

e -z . This ends the proof of the lemma.

This research was supported by the french ANR project MALIN, Projet-ANR-16-CE93-0003.

1