Limit theorem for random walk in weakly dependent random scenery

www.imstat.org/aihp 2010, Vol. 46, No. 4, 1178–1194

DOI:10.1214/09-AIHP353

© Association des Publications de l’Institut Henri Poincaré, 2010

Limit theorem for random walk in weakly dependent random scenery

Nadine Guillotin-Plantard

and Clémentine Prieur

aUniversité de Lyon, Institut Camille Jordan, Bâtiment Braconnier, 43 avenue du 11 novembre 1918, 69622 Villeurbanne Cedex, France.

E-mail:nadine.guillotin@univ-lyon1.fr

bUniversité Joseph Fourier, Tour IRMA, MOISE-LJK, B.P. 53, 38041 Grenoble Cedex 9, France. E-mail:Clementine.Prieur@imag.fr Received 12 July 2009; revised 14 August 2009; accepted 10 December 2009

Abstract. LetS=(S_k)_k≥0be a random walk onZandξ=(ξ_i)_i∈Za stationary random sequence of centered random variables, independent ofS. We consider a random walk in random scenery that is the sequence of random variables(U_n)_n≥0, where

U_n= n

k=0

ξ_Sk, n∈N.

Under a weak dependence assumption on the sceneryξ we prove a functional limit theorem generalizing Kesten and Spitzer’s [Z. Wahrsch. Verw. Gebiete50(1979) 5–25] theorem.

Résumé. SoitS=(S_k)_k≥0une marche aléatoire sur Zetξ=(ξ_i)_i∈Z une suite stationnaire de variables aléatoires centrées, indépendante deS. Nous considérons une marche aléatoire en scène aléatoire définie par la suite de variables aléatoires(Un)_n≥0= (_n

k=0ξ_Sk)_n≥0. Sous une hypothèse de dépendance faible portant sur la scèneξ, nous montrons un théorème de la limite centrale fonctionnel généralisant le théorème de Kesten et Spitzer [Z. Wahrsch. Verw. Gebiete50(1979) 5–25].

MSC:Primary 60F05; 60G50; 62D05; secondary 37C30; 37E05

Keywords:Random walks; Random scenery; Weak dependence; Limit theorem; Local time

1. Introduction

LetX=(X_i)_i≥1be a sequence of independent and identically distributed random vectors with values inZ^d. We write S0=0, Sn=

n

i=1

Xi forn≥1,

for the Z^d-random walk S=(S_n)_n∈N generated by the family X. Letξ =(ξ_x)_x∈Zd be a family of real random variables, independent of S. The sequence ξ plays the role of the random scenery. The random walk in random scenery (RWRS) is the process defined by

Un= n

k=0

ξS_k, n∈N.

RWRS was first introduced in dimension one by Kesten and Spitzer [19] and Borodin [5,6] in order to construct new self-similar stochastic processes. Functional limit theorems for RWRS were obtained under the assumption that the random variables ξ_x, x∈Z^d, are independent and identically distributed. For d =1, Kesten and Spitzer [19]

proved that when X andξ belong to the domains of attraction of different stable laws of indices 1< α≤2 and 0< β≤2, respectively, then there existsδ >¹₂such that(n^−δU_[nt])t≥0converges weakly asn→ ∞to a self-similar process with stationary increments,δ being related toαandβ byδ=1−α⁻¹+(αβ)⁻¹. The case 0< α <1 and β arbitrary is easier; they showed then that(n^−1/βU_[nt])_t≥0converges weakly, asn→ ∞, to a stable process with indexβ. Bolthausen [4] gave a method to solve the caseα=1 andβ=2 and especially, he proved that when(S_n)_n∈N is a recurrentZ²-random walk,((nlogn)^−1/2U_[nt])t≥0 satisfies a functional central limit theorem. For an arbitrary transientZ^d-random walk,n^−1/2U_nis asymptotically normal (see Spitzer [27], p. 53). Maejima [23] generalized the result of Kesten and Spitzer [19] to the case where(ξ_x)_x∈Zare i.i.d.R^d-valued random variables which belong to the domain of attraction of an operator stable random vector with exponentB. If we denote byDthe linear operator on R^d defined byD=(1−_α¹)I+_α¹B, he proved that(n^−DU_[nt])_t≥0converges weakly to an operator self similar with exponentDand having stationary increments.

One-dimensional random walks in random scenery recently arose in the study of random walks evolving on ori- ented versions ofZ²(see Guillotin-Plantard and Le Ny [15,16]) as well as in the context of charged polymers (see Chen and Khoshnevisan [8]). The understanding of these models in the case where the orientations or the charges are not independently distributed requires functional limit theorems forZ-random walk in correlated random scenery.

To our knowledge, only the case of strongly correlated stationary random sceneries has been studied by Lang and Xanh [21]. In their paper, the increments of the random walkSare assumed to belong to the domain of attraction of a non-degenerate stable law of indexα,0< α≤2. They further suppose that the sceneryξsatisfies the non-central limit theorem of Dobrushin and Major [12] with a scaling factorn^−d+(βk)/2,βk < d. Under the assumptionβk < α, it is proved that(n^{−1+βk/(2α)}U_[nt])_t≥0converges weakly asn→ +∞to a self-similar process with stationary increments, which can be represented as a multiple Wiener–Itô integral of a random function. Our aim is to study the intermediary case of a stationary random sceneryξ which satisfies a weak dependence condition introduced in Dedecker et al. [11]

and to prove Kesten and Spitzer’s theorem under this new assumption. In Guillotin-Plantard and Prieur [17] the case of a transientZ-random walk was considered and a central limit theorem for the sequence(Un)_n∈N was proved. In this paper the one-dimensional random walk will be assumed to be recurrent.

Our paper is organized as follows: In Section2, we introduce the dependence setting under which we work in the sequel. In Section3we introduce in details our model and give the main result. In Section4properties of the local time of the random walk are given as well as the ones of the intersection local time. Models for which we can compute bounds for our dependence coefficients are presented in Section5. Finally, the proof of our theorem is given in the last section.

2. Weak dependence conditions

In this section, we recall the definition of the dependence coefficients which we will use in the sequel. They have first been introduced in Dedecker et al. [11]. Our weak dependence condition will be less restrictive than the mixing one.

The reader interested in this question would find more details in Guillotin-Plantard and Prieur [17].

On the Euclidean spaceR^m, we define the metric d₁(x, y)=

m

i=1

|x_i−y_i|.

LetΛ=

m∈N^∗Λ_m, whereΛ_m is the set of Lipschitz functionsf:R^m→Rwith respect to the metric d₁. Iff ∈ Λ_m, we denote by Lip(f ):=sup_x=y^{|f (x)}_d ^{−f (y)|}

1(x,y) the Lipschitz modulus off. The set of functionsf ∈Λsuch that Lip(f )≤1 is denoted byΛ.˜

Definition 2.1. Letξbe aR^m-valued random variable defined on a probability space(Ω,A,P),assumed to be square integrable.For anyσ-algebraMofA,we define theθ2-dependence coefficient

θ2(M, ξ )=supE

f (ξ )|M

−Ef (ξ )

2, f∈ ˜Λ . (2.1)

Here, X ₂= [E(X²)]^1/2denotes the norm inL².

We now define the coefficientθ_k,2for a sequence ofσ-algebras and a sequence ofR-valued random variables.

Definition 2.2. Let(ξ_i)_i∈Z be a sequence of square integrable random variables valued in R.Let (Mi)_i∈Z be a sequence ofσ-algebras ofA.For anyk∈N^∗∪ {∞}andn∈N,we define

θk,2(n)= max

1≤l≤k

1 l sup

θ2

Mp, (ξj₁, . . . , ξj_l)

, p+n≤j1<· · ·< jl

and

θ₂(n)=θ_∞,2(n)=sup

k∈N^∗θ_k,2(n).

Definition 2.3. Let(ξ_i)_i∈Z be a sequence of square integrable random variables valued in R.Let (Mi)_i∈Z be a sequence ofσ-algebras ofA.The sequence (ξ_i)_i∈Z is said to beθ₂-weakly dependent with respect to(Mi)_i∈Z if θ₂(n)−→n→+∞0.

Remark. Replacing the · 2norm in(2.1)by the · 1norm,we get theθ1dependence coefficient first introduced by Doukhan and Louhichi[14].

3. Model and results

LetS=(S_k)_k≥0be a Z-random walk (S0=0) whose increments (X_i)_i≥1 are centered and square integrable. We denote byPX₁ the law of the random variableX₁. For anyq∈N^∗such thatP(X1∈ [−q, q])is non zero, we define the probability measure onZ

Pq= PX1|_[−q,q]

P(X1∈ [−q, q]).

The random walkSis said to satisfy the property(P)if there existsq∈N^∗such that x∈Z; ∃n,P⁽ⁿ⁾_q (x) >0 =Z.

In particular, if there exists someq∈N^∗such that the random walk associated toPqis centered and aperiodic thenS satisfies the property(P). For instance, the simple random walk onZverifies(P). Letξ=(ξ_i)_i∈Zbe a sequence of centered real random variables. The sequencesSandξ are defined on a same probability space denoted by(Ω,F,P) and are assumed to be independent. We are interested in the asymptotic behaviour of the following sum

U_n= n

k=0

ξ_Sk.

The case where theξi’s are independent and identically distributed random variables with positive variance has been considered by Kesten and Spitzer [19] and Borodin [5,6]. They studied the weak convergence of the sequence of stochastic processes

n^−3/4U_nt, t≥0, n≥1,

whereU_s is defined as the linear interpolation

Us=Un+(s−n)(Un+1−Un) whenn≤s≤n+1.

Consider a standard Brownian motion(B_t)_t≥0, denote by(L_t(x))_t≥0its corresponding local time atx∈Rand intro- duce a pair of independent Brownian motions(Z₊(x), Z₋(x)), x≥0 defined on the same probability space as(Bt)t≥0

and independent of him. The following process is well-defined for allt≥0:

Δ_t= _∞

0

L_t(x)dZ₊(x)+ _∞

0

L_t(−x)dZ₋(x). (3.1)

It was proved by Kesten and Spitzer [19] that this process has a self-similar continuous version of index ³₄, with stationary increments. We denote by⇒^C the convergence in the space of continuous functionsC([0,∞),R)endowed with the uniform topology.

Theorem 3.1 (Kesten and Spitzer [19]). Assume that theξ_i’s are independent and identically distributed with posi- tive varianceσ²>0.Then,

1 n^3/4Unt

t≥0

⇒C (σ Δt)t≥0. (3.2)

A simple proof of this theorem was proposed by Cadre [7], Section 2.5.a., using a weak limit theorem for stochastic integrals (Theorem 1.1. in Kurtz and Protter [20]). Applying Cadre’s method, Theorem3.1can be extended to any scenery given by a stationary and ergodic sequence of square integrable martingale differences. Then, a natural idea is to generalize the result to any stationary and ergodic sequenceξ of square integrable random variables as it was done for the central limit theorem. Under suitable assumptions on the sequence, for instance the convergence of the series_∞

k=0E(ξ_k|M0)inL², the sceneryξ is equal to a martingale differences sequence modulo a coboundary term and satisfies a Donsker theorem. However, the RWRS associated to the coboundary term (if it is non zero) is not negligible. It can be proved that the L²-norm of this sum correctly normalized byn^3/4 converges to some positive constant.

In order to weaken the assumptions on the fieldξwe introduce a sequence(Mi)_i∈Zofσ-algebras ofFdefined by Mi=σ (ξj, j≤i), i∈Z.

In the sequel, the dependence coefficients will be defined with respect to the sequence ofσ-algebras(Mi)_i∈Z. Theorem 3.2. Assume that the following conditions are satisfied:

(A0) The random walkSsatisfies the property(P).

(A1) ξ= {ξ_i}i∈Zis a stationary sequence of square integrable random variables.

(A2) θ₂^ξ(·)is bounded above by a non-negative functiong(·)such that:

• x→x^3/2g(x)is non-increasing,

• ∃0< ε <1,_∞

i=02^3i/2g(2^iε) <∞.

Then,asntends to infinity, 1

n^3/4Unt

t≥0

⇒C

i∈Z

E(ξ0ξi)(Δt)t≥0. (3.3)

Remark. Assumptions(A1)and(A2)imply that

∀λ∈ [0,1/2[,

k∈Z

|k|^λE(ξ0ξk)<+∞. (3.4)

Indeed,this sum is equal to E(ξ₀²)+2

∞ k=1

|k|^λE(ξ0ξk)

and for anyk≥1,from Cauchy–Schwarz inequality,we get E(ξ₀ξ_k)=E

ξ₀E(ξ_k|M0)

≤ ξ_{0 2}θ₂^ξ(k)

≤ ξ0 2g(k).

The result(3.4)follows by remarking that ∞

k=1

|k|^λg(k)≤g(1) ∞ k=1

1 k^3/2−λ which is finite for everyλ∈ [0,1/2[.

Remark. Ifθ₂^ξ(n)=O(n^−a)for some positivea,condition(A2)holds fora >3/2.

4. Properties of the occupation times of the random walk

The random walkS=(Sk)k≥0is defined as in the previous section and is assumed to verify the property(P). The local time of the random walk is defined for everyi∈Zby

N_n(i)= n

k=0

1_{S_k=i}.

The local time of self-intersection at pointiof the random walk(S_n)_n≥0is defined by α(n, i)=

n

k,l=0

1_{Sk−Sl=i}.

The stochastic properties of the sequences (N_n(i))_n∈N,i∈Z and(α(n, i))_n∈N,i∈Z are well-known when the random walk S is strongly aperiodic. A random walk who satisfies the property (P) is not strongly aperiodic in general.

However, a local limit theorem for the random walks satisfing(P)was proved by Cadre [7] (see Lemma 2.4.5, p. 70), then it is not difficult to adapt the proofs of the strongly aperiodic case to our setting: for assertion (i) see Lemma 4 in Kesten and Spitzer [19], for (ii)(a) see Lemma 3.1 in Dombry and Guillotin-Plantard [13]. Result (ii)(b) is obtained from Lemma 6 in Kesten and Spitzer [19]; details are omitted. Assertion (iii) is an adaptation to dimension one of Lemma 2.3.2 of Cadre’s [7] thesis.

Proposition 4.1. (i)The sequencen^−3/4maxi∈ZN_n(i)converges in probability to0.

(ii) (a) For anyp∈ [1,+∞),there exists some constantCsuch that for alln≥1, E

α(n,0)^p

≤Cn^3p/2.

(b) For anym≥1,for any realθ₁, . . . , θ_m,for any0≤t₁≤ · · · ≤t_m,the sequence n^−3/2

i∈Z

_m

k=1

θ_kN_[ntk](i) 2

converges in distribution to

R

_m

k=1

θkLt_k(x) 2

dx,

where(Lt(x))t≥0;x∈Ris the local time of the real Brownian motion(Bt)t≥0.

(iii) For everyλ∈]0,1[,there exists a constantCsuch that for anyi, j∈Z, α(n, i)−α(n, j )

2≤Cn^(3−λ)/2|i−j|^λ.

5. Examples

In this section, we present examples for which we can compute upper bounds forθ2(n)for anyn≥1. We refer to Chapter 3 in Dedecker et al. [11] and references therein for more details.

5.1. Example1:Causal functions of stationary sequences

Let (E,E,Q) be a probability space. Let (ε_i)_i∈Z be a stationary sequence of random variables with values in a measurable space S. Assume that there exists a real valued function H defined on a subset of S^N, such that H (ε₀, ε₋₁, ε₋₂, . . .)is defined almost surely. The stationary sequence(ξ_n)_n∈Zdefined byξ_n=H (ε_n, ε_n−1, ε_n−2, . . .) is called a causal function of(εi)i∈Z.

Assume that there exists a stationary sequence(ε_i)_i∈Zdistributed as(ε_i)_i∈Z and independent of(ε_i)_i≤0. Define ξ_n^∗=H (ε_n, ε_n−1, ε_n−2, . . .). Clearly,ξ_n^∗is independent ofM0=σ (ξ_i, i≤0)and distributed asξ_n. Let(δ₂(i))_i>0be a non increasing sequence such that

Eξ_i−ξ_i^∗|M0

2≤δ₂(i). (5.1)

Then the coefficientθ₂of the sequence(ξ_n)_n≥0satisfies

θ₂(i)≤δ₂(i). (5.2)

Let us consider the particular case where the sequence of innovations(ε_i)_i∈Z is absolutely regular in the sense of Volkonskii and Rozanov [26]. Then, according to Theorem 4.4.7 in Berbee [2], ifE is rich enough, there exists (ε_i)_i∈Zdistributed as(εi)_i∈Zand independent of(εi)i≤0such that

Q

ε_i=ε_ifor somei≥k|F0

=1

2 Qε_˜k|F0−Qε_˜k v,

whereε˜_k=(ε_k, ε_k+1, . . .),F0=σ (ε_i, i≤0)and · vis the variation norm. In particular if the sequence(ε_i)_i∈Zis idependent and identically distributed, it suffices to takeε_i=ε_i fori >0 andε_i−ε_ifori≤0, where(ε_i)_i∈Zis an independent copy of(ε_i)_i∈Z.

Application to causal linear processes In that case,ξ_n=

j≥0a_jε_n−j, where(a_j)_j≥0is a sequence of real numbers. We can choose δ2(i)≥ε0−ε₀

2

j≥i

|aj| +

i−1

j=0

|aj|εi−j−ε_i−j

2.

From Proposition 2.3 in Merlevède and Peligrad [24], we obtain that

δ₂(i)≤ε₀−ε₀

2

j≥i

|a_j| +

i−1

j=0

|a_j|

2²

_{β(σ (εk,k≤0),σ (εk,k≥i−j ))}

0

Q²_ε

0(u) 1/2

du,

whereQε₀is the generalized inverse of the tail functionx→Q(|ε0|> x).

5.2. Example2:Iterated random functions

Let(ξn)n≥0be a real valued stationary Markov chain, such thatξn=F (ξn−1, εn)for some measurable functionF and some independent and identically distributed sequence(εi)i>0 independent ofξ0. Letξ₀^∗ be a random variable distributed asξ0and independent of(ξ0, (εi)i>0). Defineξ_n^∗=F (ξ_n^∗₋₁, εn).The sequence(ξ_n^∗)n≥0is distributed as (ξn)n≥0and independent ofξ0. LetMi=σ (ξj,0≤j≤i). As in Example 1, define the sequence(δ2(i))i>0by (5.1).

The coefficientθ2of the sequence(ξn)n≥0satisfies the bound (5.2) of Example 1.

Letμbe the distribution ofξ0and(ξ_n^x)n≥0be the chain starting fromξ₀^x=x. With these notations, we can choose δ2(i)such that

δ₂(i)≥ξ_i−ξ_i^∗

2= ξ_i^x−ξ_i^y2

2μ(dx)μ(dy) 1/2

.

For instance, if there exists a sequence(d₂(i))_i≥0of positive numbers such that ξ_i^x−ξ_i^y

2≤d2(i)|x−y|,

then we can takeδ₂(i)=d₂(i) ξ₀−ξ₀^∗ ₂. For example, in the usual case where F (x, ε₀)−F (y, ε₀) ₂≤κ|x−y| for someκ <1, we can taked₂(i)=κⁱ.

An important example isξ_n=f (ξ_n−1)+ε_nfor someκ-Lipschitz functionf. Ifξ₀has a moment of order 2, then δ₂(i)≤κⁱ ξ₀−ξ₀^∗ ₂.

5.3. Example3:Dynamical systems on[0,1]

LetI = [0,1],T be a map fromI toI and defineX_i =Tⁱ. Ifμis invariant byT, the sequence(X_i)_i≥0of random variables from(I, μ)toI is strictly stationary.

For any finite measureνonI, we use the notationsν(h)=

Ih(x)ν(dx). For any finite signed measureνonI, let ν = |ν|(I )be the total variation ofν. Denote by g _1,λtheL¹-norm with respect to the Lebesgue measureλonI. Covariance inequalities

In many interesting cases, one can prove that, for anyBVfunctionhand anykinL¹(I, μ), cov

h(X₀), k(X_n)≤a_nk(X_n)

1

h _1,λ+ dh

, (5.3)

for some nonincreasing sequencea_ntending to zero asntends to infinity.

Spectral gap

Define the operatorLfromL¹(I, λ)toL¹(I, λ)via the equality 1

0

L(h)(x)k(x)dλ(x)= 1

0

h(x)(k◦T )(x)dλ(x), whereh∈L¹(I, λ)andk∈L^∞(I, λ).

The operatorLis called the Perron–Frobenius operator ofT. In many interesting cases, the spectral analysis ofLin the Banach space ofBV-functions equiped with the norm h _v= dh + h _1,λcan be done by using the Theorem of Ionescu-Tulcea and Marinescu (see Lasota and Yorke [22] and Hofbauer and Keller [18]). Assume that 1 is a simple eigenvalue ofLand that the rest of the spectrum is contained in a closed disk of radius strictly smaller than one. Then there exists a uniqueT-invariant absolutely continuous probabilityμwhose densityf_μisBV, and

Lⁿ(h)=λ(h)fμ+Ψⁿ(h) withΨⁿ(h)

v≤Kρⁿ h v (5.4)

for some 0≤ρ <1 andK >0. Assume moreover that

I_∗= {fμ=0}is an interval, and there existsγ >0 such thatfμ> γ⁻¹onI_∗. (5.5)

Without loss of generality assume thatI_∗=I (otherwise, take the restriction toI_∗in what follows). Define now the Markov kernel associated toT by

P (h)(x)=L(f_μh)(x)

f_μ(x) . (5.6)

It is easy to check (see for instance Barbour, Gerrard and Reinert [1]) that(X0, X1, . . . , X_n)has the same distribution as(Yn, Yn−1, . . . , Y0)where(Yi)i≥0is a stationary Markov chain with invariant distributionμand transition kernel P. Since f g _∞≤ f g v≤2 f v g v, we infer that, takingC=2Kγ ( dfμ +1),

Pⁿ(h)=μ(h)+g_n with g_{n ∞}≤Cρⁿ h _v. (5.7)

This estimate implies (5.3) witha_n=Cρⁿ(see Dedecker and Prieur [10]).

Expanding maps

Let([ai, ai+1[)1≤i≤N be a finite partition of[0,1[. We make the same assumptions onT as in Collet, Martinez and Schmitt [9]:

1. For each 1≤j≤N, the restrictionTj ofT to]aj, aj+1[is strictly monotonic and can be extented to a function T_j belonging toC²([a_j, a_j+1]).

2. LetI_nbe the set where(Tⁿ)is defined. There existsA >0 ands >1 such that infx∈In|(Tⁿ)(x)|> Asⁿ.

3. The mapT is topologically mixing: for any two nonempty open setsU, V, there existsn₀≥1 such thatT⁻ⁿ(U )∩ V =∅for alln≥n₀.

If T satisfies 1., 2. and 3., then (5.4) holds. Assume furthermore that (5.5) holds (see Morita [25] for sufficient conditions). Then, arguing as in Example 4 in Section 7 of Dedecker and Prieur [10], we can prove that for the Markov chain(Y_i)_i≥0and theσ-algebrasMi=σ (Y_j, j≤i), there exists a positive constantCsuch thatθ₂(i)≤Cρⁱ. 6. Proof of Theorem3.2

6.1. Notations and preliminary technical lemmas

For anyn∈Nand anyi∈Z, we denote byX_n,i the random variableN_n(i)ξ_i whereN_n(i)denotes the local time of the random walk at sitei.

We denote byG=σ (S_k, k≥0)theσ-algebra generated by the random walkS. For any real random variables X, Y, and anyt∈R, we define

dt(X, Y )=E e^{it X}|G

−E

e^{it Y}|G.

Letηbe a random variable with standard normal distribution, independent of the random walkS. Let(X_i)_i=1,...,3be random variables such that fori=1, . . . ,3,E(X_i|G)=0 andE(X²_i|G)is bounded. LetY1, Y2be random variables such thatE(Yi|G)=0 andE(Y_i²|G)is bounded fori=1,2. They are assumed to be independent conditionally to the random walk. LetXbe a real random variable and lett∈R, we define

At(X)=dt

X, η E

X²|G . The following properties ofd_t andA_t hold.

Lemma 6.1 (Lemma 4.3 in Utev [28]).

A_t(X₁)≤2 3|t|³E

|X₁|³|G , At(Y1+Y2)≤At(Y1)+At(Y2),

dt(X2+X3, X2)≤t² 2

E X₃²|G

+ E

X₂²|G E

X²₃|G1/2 ,

∀a, b∈R, d_t(ηa, ηb)≤t²

2a²−b².

The following lemma is a variation on Lemma 1.2. in Utev [28]; the proof is omitted.

Lemma 6.2. For everyε∈ ]0,1[,denoteδ_ε=(1−ε²+2ε)/2.Let a non-decreasing sequence of non-negative num- bersa(n) be specified, such that there exist non-increasing sequences of non-negative numbersε(k),γ (k) and a sequence of naturalsT (k),satisfying conditions

T (k)≤2⁻¹ k+

k^δε , a(k)≤ max

k₀≤s≤k

a T (s)

+γ (s)

for anyk≥k₀with an arbitraryk₀∈N^∗.Then a(n)≤a(n₀)+2

k₀≤2^j≤n

γ 2^j

for anyn≥k₀,where one can taken₀=2^cwithc >²₁⁻₋^δ_δ^ε

ε. For anyM >0, we define the functions

ϕM:

R→R,

x→ϕM(x)=(x∧M)∨(−M) and

ϕ^M:

R→R,

x→ϕ^M(x)=x−ϕ_M(x).

Let(ε_n)_n≥1be a sequence of positive numbers tending to 0 asngoes to infinity. For anyn∈Nand anyi∈Z, we denote byZ_n,ithe random variable

ϕ_εnn3/4(Xn,i)−E

ϕ_εnn3/4(Xn,i)|G .

The next lemma will be a key point in the proof of Theorem3.2.

Lemma 6.3. Let0< ε <1.There exists some positive constantC(ε)such that for alla∈Z,for allv∈N^∗, A_t

n^−3/4

a+v

i=a+1

Z_n,i

is bounded by C(ε)

|t|³h^2/εn^−9/4

a+v

i=a+1

E

|Z_n,i|³|G +t²

h^(ε−1)/2+

j:2^j≥h^1/ε

2^3j/2g 2^{j ε}

n^−3/2

a+v

i=a+1

N_n(i)²

,

wherehis an arbitrary positive natural number andgis the function introduced in assumption(A2)of Theorem3.2.

Proof. Leth∈N^∗. Let 0< ε <1. In the following,C,C(ε)denote constants which may vary from line to line. Let κεbe a positive constant greater than 1 which will be precised further.

Case 1: v < κ_εh^1/ε. By Lemma6.1, we have

A_t

n^−3/4

a+v

i=a+1

Z_n,i

≤ 2|t|³ 3n^9/4E

a+v

i=a+1

Z_n,i

3G

≤ 2

3n^9/4κ_ε²|t|³h^2/ε

a+v

i=a+1

E

|Z_n,i|³|G

(6.1)

since|x|³is a convex function.

Case 2: v≥κεh^1/ε.

Without loss of generality, assume thata=0. Letδε=(1−ε²+2ε)/2. Define then m= [v^ε], B=

u∈N: 2⁻¹ v−

v^δε

≤um≤2⁻¹v , A=

u∈N: 0≤u≤v,

(u+1)m i=um+1

N_n(i)²≤(m/v)^ε v

i=1

N_n(i)²

.

Following Utev [28] we prove that, for 0< ε <1,A∩Bis not wide forvgreater thanκε. We have indeed

|A∩B| = |B| − |A∩B| ≥ |B| − |A| ≥v^(1−ε2)/2 2

1−4v^{−(1−ε)2/2}

−3 2,

whereAdenotes the complementary of the setA. We can findκεlarge enough so that|A∩B|be positive.

Letu∈A∩B. We start from the following simple identity Q≡n^−3/4

v

i=1

Zn,i

=n^−3/4 um

i=1

Z_n,i+n^−3/4

(u+1)m i=um+1

Z_n,i+n^−3/4 v

i=(u+1)m+1

Z_n,i

≡Q₁+Q₂+Q₃. (6.2)

For any fixedn≥0, and anyi∈Zsuch thatN_n(i)=0, define W_n,i=ϕ_ε

nn^3/4/N_n(i)(ξ_i)−E ϕ_ε

nn^3/4/N_n(i)(ξ_i)|G .

IfN_n(i)=0, letW_n,i=0. As for any fixedn≥0,i∈Z, the function x→ϕ_εnσn/Nn(i)(x)−E

ϕ_εnσn/Nn(i)(ξ_i)|G

is 1-Lipschitz, we have for any fixed path of the random walk, for alll≥1, for allk≥1, θ_k,2^W·,n(l)≤θ_k,2^ξ (l),

whereW_·,n=(W_n,i)_i∈Zandξ=(ξ_i)_i∈Z. We now claim that for any fixedn, anya, b∈N, E

_b

i=a

Zn,i

2

G

≤C b

i=a

Nn(i)² (6.3)

withC=2E(ξ₀²)+2√

2E(ξ₀²)^1/2_∞

l=1θ_1,2(l)which is finite from assumptions(A1)and(A2). Indeed, remark that Z_n,i=N_n(i)W_n,i, then

E _b

i=a

Z_n,i 2

G

=E _b

i=a

N_n(i)W_n,i 2

G

= b

i=a

N_n(i)²E W_n,i² |G

+ b

i=a

j∈{a,...,b};j=i

N_n(i)N_n(j )E(W_n,iW_n,j|G)

≤ b

i=a

N_n(i)²E W_n,i² |G

+ b

i=a

N_n(i)²

j∈{a,...,b};j=i

E(W_n,iW_n,j|G) by remarking thatN_n(i)N_n(j )≤¹₂(N_n(i)²+N_n(j )²).

Then for anyj > i, using Cauchy–Schwarz inequality, we obtain that E(W_n,iW_n,j|G)=E

W_n,iE(W_n,j|Mi)|G

≤E

W_n,i² |G1/2

E

E(W_n,j|Mi)²|G1/2

≤E

W_n,i² |G1/2

θ_1,2^ξ (j−i).

By remarking thatE(W_n,i² |G)≤E(ξ₀²)(since|ϕ_M(x)| ≤ |x|for any realx), we deduce inequality (6.3).

By Lemma6.1,

d_t(Q, Q₁+Q₃)=d_t(Q, Q−Q₂)≤t² 2

E Q²₂|G

+E

Q²₂|G1/2

E

Q²|G1/2

. (6.4)

Using (6.4) and (6.3), we get

dt(Q, Q1+Q3)≤Ct²v^(ε−1)ε/2n^−3/2 v

i=1

Nn(i)². (6.5)

Now, given the random variablesQ₁andQ₃, we define two random variablesg₁andg₃which are assumed indepen- dent conditionally to the random walkSsuch that conditionally to the random walk, the distribution ofg_i coincides with that ofQ_i,i=1,3. We have

d_t(Q₁+Q₃, g₁+g₃)

=E

e^{it Q1}−1

e^{it Q3}−1

|G

−E

e^{it Q1}−1|G E

e^{it Q3}−1|G

≤Ee^{it Q1}−1²|G1/2

EE

e^{it Q3}−1−E

e^{it Q3}−1|G

|Mum;G²|G

≤2|t|n^−3/4E

um

i=1

Z_n,i

2G 1/2

v|t|n^−3/4 _v

i=(u+1)m+1

N_n(i)

θ₂^ξ(m+1)

≤Ct²v^3/2n^−3/2 _v

i=1

N_n(i)²

g v^ε

by (6.3), Definition2.2and assumption(A2)of Theorem3.2. Hence, dt(Q1+Q3, g1+g3)≤Ct²f (v)n^−3/2

v

i=1

Nn(i)², (6.6)