Fabienne Castell
email: fabienne.castell@cmi.univ-mrs.fr

FRANC ¸OISE P ÈNE Nadine Guillotin-Plantard

Bruno Schapira
email: bruno.schapira@math.u-psud.fr

A LOCAL LIMIT THEOREM FOR RANDOM WALKS IN RANDOM SCENERY AND ON RANDOMLY ORIENTED LATTICES

Keywords: 2000 Mathematics Subject Classification. 60F05, 60G52 Random walk in random scenery, random walk on randomly oriented lattices, local limit theorem, stable process

Random walks in random scenery are processes defined by Zn := n k=1 ξX 1 +...+X k , where (X k , k ≥ 1) and (ξy, y ∈ Z) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2] respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α = 1 and as n → ∞, of n -δ Zn, for some suitable δ > 0 depending on α and β. Here we are interested in the convergence, as n → ∞, of n δ P(Zn = ⌊n δ x⌋), when x ∈ R is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

1. Introduction 1.1. About the model. Random walks in random scenery (RWRS) are simple models of processes in disordered media with long-range correlations. They have been used in a wide variety of models in physics to study anomalous dispersion in layered random flows [START_REF] Matheron | Is transport in porous media always diffusive? A counterxample[END_REF], diffusion with random sources, or spin depolarization in random fields (we refer the reader to Le Doussal's review paper [START_REF] Doussal | Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields[END_REF] for a discussion of these models).

On the mathematical side, motivated by the construction of new self-similar processes with stationary increments, Kesten and Spitzer [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] and Borodin [START_REF] Borodin | A limit theorem for sums of independent random variables defined on a recurrent random walk. (Russian)[END_REF][START_REF] Borodin | Limit theorems for sums of independent random variables defined on a transient random walk. Investigations in the theory of probability distributions[END_REF] introduced RWRS in dimension one and proved functional limit theorems. These processes are defined as follows. Let ξ := (ξ y , y ∈ Z) and X := (X k , k ≥ 1) be two independent sequences of independent identically distributed random variables taking values in R and Z respectively. The sequence ξ is called the random scenery. The sequence X is the sequence of increments of the random walk (S n , n ≥ 0) defined by S 0 := 0 and S n := n i=1 X i , for n ≥ 1. The random walk in random scenery Z is then defined for all n ≥ 1 by

Z n := n-1 k=0 ξ S k .
Denoting by N n (y) the local time of the random walk S :

N n (y) = #{k = 0, ..., n -1 : S k = y} , it is straightforward to see that Z n can be rewritten as Z n = y ξ y N n (y).

As in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF], the distribution of ξ 1 is assumed to belong to the normal domain of attraction of a strictly stable distribution S β of index β ∈ (0, 2], with characteristic function given by φ(u) = e -|u| β (A 1 +iA 2 sgn(u)) u ∈ R, where 0 < A 1 < ∞ and |A -1 1 A 2 | ≤ | tan(πβ/2)|. When β = 1, this is the most general form of a strictly stable distribution. In the case β = 1, this is the general form of a random variable Y with strictly stable distribution satisfying the following symmetry condition :

sup M >0 |E(Y 1 {|Y |<M } )| < +∞.
(1)

We will denote by f β the density function of the law S β .

Concerning the random walk, the distribution of X 1 is assumed to belong to the normal domain of attraction of a strictly stable distribution S α with index α ∈ (0, 2]. In this paper we will actually not consider the case α = 1 (see Remark 2 in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] for some discussion on this case).

Then the following weak convergences hold in the space of càd-làg real-valued functions defined on [0, ∞) and on R respectively :

n -1 α S ⌊nt⌋ t≥0 L =⇒ n→∞ (U (t)) t≥0
and

  n -1 β ⌊nx⌋ k=0 ξ k   x∈R L =⇒ n→∞ (Y (x)) x∈R ,
where U and Y are two independent Lévy processes such that U (0) = 0, Y (0) = 0, U (1) has distribution S α , Y (1) and Y (-1) have distribution S β . When α ∈ (1, 2], the random walk (S n , n ≥ 0) is recurrent, and the limiting process U admits a local time process. We denote by (L t (x), t ∈ R + , x ∈ R) the jointly continuous version of this local time.

Let

δ := 1 - 1 α + 1 αβ .
Papers [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF][START_REF] Borodin | A limit theorem for sums of independent random variables defined on a recurrent random walk. (Russian)[END_REF][START_REF] Borodin | Limit theorems for sums of independent random variables defined on a transient random walk. Investigations in the theory of probability distributions[END_REF] proved that the following weak convergences hold in the space of continuous real-valued functions defined on [0, ∞) :

if α > 1, n -δ Z nt t≥0 L =⇒ n→∞ (∆(t)) t≥0 (2)
if α < 1, n

-1 β Z nt t≥0 L =⇒ n→∞ Y (t) E[( N β-1 ∞ (0)] 1 β t≥0 , (3) where
• Z s is defined as the linear interpolation

Z s = Z n +(s-n)(Z n+1 -Z n ) when n ≤ s ≤ n+1,
• ∆ is the process defined by

∆(t) = +∞ -∞ L t (x) dY (x) ,
• N ∞ (0) is the total time spent in 0 by the two-sided random walk (S k , k ∈ Z) with S -k = -k m=1 X -m (where (X -k , k ≥ 1) is independent of (X k , k ≥ 1) and with the same distribution).

The limiting process ∆ is known to be a continuous δ-self-similar process with stationary increments. It can be seen as a mixture of β-stable processes, but it is not a stable process.

Since these seminal papers, RWRS have been extensively studied. Far from being exhaustive, we can cite limit theorems in higher dimension [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF], strong approximation results and laws of the iterated logarithm [START_REF] Khoshnevisan | A law of the iterated logarithm for stable processes in random scenery[END_REF][START_REF] Csáki | An embedding for the Kesten-Spitzer random walk in random scenery[END_REF][START_REF] Csáki | Strong invariance for local times Z. Wahrscheinlichkeitstheorie verw[END_REF], limit theorems for correlated sceneries or walks [START_REF] Guillotin-Plantard | Limit theorem for random walk in weakly dependent random scenery[END_REF][START_REF] Cohen | Convergence of dependent walks in a random scenery to fBm-local time fractionnal stable motions[END_REF], large and moderate deviations results [START_REF] Castell | Annealed large deviations for diffusions in a random Gaussian shear flow drift[END_REF][START_REF] Castell | Moderate deviations for diffusions in a random Gaussian shear flow drift[END_REF][START_REF] Asselah | Random walk in random scenery and self-intersection local times in dimensions d ≥ 5[END_REF][START_REF] Gantert | Annealed deviations of random walk in random scenery[END_REF]. Our contribution in this paper is a local version of the convergence results from [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF], as we make more precise in the next subsection. Our first result concerns the case α > 1 :

Theorem 1. Lattice case, α > 1.

Assume that α ∈ (1, 2] and β ∈ (0, 2]. Let C(x) be the continuous function defined by

C(x) := E |L| -1 β f β (|L| -1 β x) for all x ∈ R,
where

|L| β := R L β 1 (y) dy 1/β
. Then, for every x ∈ R, we have 0 < C(x) < ∞ and

• if P nξ 1 -n δ x / ∈ dZ = 1, then P Z n = n δ x = 0;
• if P nξ 1 -n δ x ∈ dZ = 1, then

P Z n = n δ x = d C(x) n δ + o(n -δ ) ,
where the o(n -δ ) is uniform in x.

Remark. There is no other alternative for the law of ξ 1 . Indeed, let b be in the support of ξ 1 . Then nξ 1 belongs to nb + dZ. Hence the condition nξ 1 -⌊n δ x⌋ ∈ dZ is equivalent to ⌊n δ x⌋ -nb ∈ dZ.

Our second result concerns the case α < 1 :

Theorem 2. Lattice case, α < 1.
Assume that α ∈ (0, 1), β ∈ (0, 2] and x ∈ R. Let D(x) := rf β (rx), with r

:= E[ N β-1 ∞ (0)] -1/β . Then • if P nξ 1 -n 1 β x / ∈ dZ = 1, then P Z n = n 1 β x = 0; • if P nξ 1 -n 1 β x ∈ dZ = 1, then P Z n = n 1 β x = d D(x) n 1 β + o(n -1 β ) ,
where the o(n

-1 β ) is uniform in x;
Finally we get the local limit theorem when ξ is strongly nonlattice, i.e. when lim sup

|u|→+∞ |ϕ ξ (u)| < 1.
Theorem 3. Strongly nonlattice case.

• If α > 1 and β ∈ (0, 2], then for all a, b ∈ R such that a < b,

lim n→∞ n δ P Z n ∈ [n δ x + a; n δ x + b] = C(x)(b -a) .
• If α < 1 and β ∈ (0, 2], then for all a, b ∈ R such that a < b,

lim n→∞ n 1 β P Z n ∈ [n 1 β x + a; n 1 β x + b] = D(x)(b -a) .
1 Note that ξ is said to be non-arithmetic if d = 1.

On the one hand, these results give some qualitative information about the behaviour of Z. For instance the transience of the process Z is easily deduced (with Borel-Cantelli Lemma) when β < 1. Note that since Z is not a Markov chain, the recurrence property when β > 1 does not directly follow from the above local limit theorems. However this can be proved by using an argument from ergodic theory (see [START_REF] Schmidt | On recurrence[END_REF]). Indeed, it is enough to remark that when β ∈ (1, 2], the random variables ξ S k , k ∈ N form an ergodic and stationary sequence of integrable and centered random variables.

On the other hand this work was motivated by the study of random walks on randomly oriented lattices. In the simplest case, one should think to the simple random walk defined on a random sublattice of the oriented lattice Z 2 , which is constructed as follows. On each horizontal line, one removes all edges oriented to the right with probability 1/2 or those oriented to the left with probability 1/2, and so independently on each level. Then it is known, and not difficult to see, that the first coordinate of the resulting random walk is closely related to a random walk in random scenery Z = k ξ S k , with S the simple random walk on Z and the ξ y i.i.d random variables with geometric distribution (see Section 5 or [START_REF] Guillotin-Plantard | A functional limit theorem for a 2d-random walk with dependent marginals[END_REF] for more explanations). In [START_REF] Guillotin-Plantard | A functional limit theorem for a 2d-random walk with dependent marginals[END_REF] it was conjectured that the probability of return to the origin of this random walk is equivalent to a constant times n -5/4 . Here we prove a local limit theorem for even more general random walks, giving in particular a proof of this conjecture. We refer the reader to Section 5 for more precise statements of our results.

1.3. Outline of the proof. Let us give a very rough description of the proofs for RWRS. To fix ideas, we do it for x = 0 and α > 1. By Fourier inverse transform, we have to study the asymptotic behavior of

E e itZn dt = E   y∈Z Z ϕ ξ (tN n (y))   dt . (4)
For t such that tN n (y) is small, only the behavior of ϕ ξ around 0 is relevant. Therefore, for |t| ≤ (sup y N n (y)

) -1 ≃ n -1+1/α , E   y∈Z Z ϕ ξ (tN n (y))   ≃ E exp(-|t| β y N n (y) β (A 1 + iA 2 sgn(t))) .
Now, y N n (y) β is of order n βδ , and a change of variable t n δ t leads to the dominant part in the integral (4).

For t ≥ (sup y N n (y)) -1 ≃ n -1+1/α , the behavior of ϕ ξ away from 0 comes into play. In the strongly nonlattice case, one can find ǫ 0 > 0 and ρ ∈ (0, 1) such that

|ϕ ξ (t)| ≤ ρ for |t| ≥ ǫ 0 , so that for |t| ≥ n -1+1/α , y∈Z Z ϕ ξ (tN n (y)) ≤ ρ #{y;Nn(y)≥ ǫ 0 t } ≤ ρ #{y;Nn(y)≥ǫ 0 n 1-1/α } .
It is easily seen that there is a large number of points visited at least n 1-1/α times, leading to the result.

The lattice case is more delicate, since in this case |ϕ ξ (t)| = 1 for t ∈ 2π d Z, so that the inequality |ϕ ξ (tN n (y))| ≤ ρ is only valid for the y such that d(tN n (y); 2π d Z) ≥ ǫ 0 . Thus, the main difficulty is to show that for |t| ≥ n 1-1/α , there are a lot of such sites. This is done by a surgery on the trajectories of the random walk.

Let us briefly describe now the organization of the paper. In the next section, we prove Theorem 1. In Sections 3 and 4, we sketch the proofs of Theorem 2 and Theorem 3 which are easier and follow the same lines. In Section 5, the local limit theorem for random walks evolving on randomly oriented lattices is obtained by using similar techniques as for the proof of Theorem 1. Finally in the appendix, we prove some auxiliary results on the range of the random walk S, that we should need, but which could also be of independent interest.

Lattice case

, α > 1: Proof of Theorem 1 2.1. Finiteness of C(x). Lemma 4. For all x ∈ R, 0 < C(x) < +∞. Proof. Let x ∈ R. Since R L 1 (y) dy = 1 and β ≤ 2, we have a.s. R L β 1 (y) dy ≤ 1 + sup y L 1 (y) (β-1) + . Hence R L β 1 (y) dy is a.s. finite. So C(x) > 0. Let us prove now that C(x) is finite. First we have C(x) ≤ f β ∞ E[|L| -1 β ].
Let us assume now that β > 1. By Hölder's inequality,

1 = R L 1 (y) dy ≤ |L| β R 1(L 1 (y) > 0) dy 1-1 β .
Thus by using Jensen's inequality we get

C(x) ≤ f β ∞ E R 1(L 1 (y) > 0) dy 1-1 β ≤ f β ∞ E R 1(L 1 (y) > 0) dy 1-1 β = f β ∞ (E[λ(U ([0, 1]))]) 1-1 β ,
where λ denotes the Lebesgue measure on R and U ([0, 1]) the set of points visited by U before time 1. This finishes the proof in the case β > 1, since the last quantity is finite (see for example [START_REF] Gall | The range of stable random walks[END_REF] p.703).

Next, if β = 1, then |L| β = 1 and C(x) = f β (x) < +∞. Assume finally that β < 1. Then 1 = R L 1 (y) dy ≤ |L| β β sup x L 1 (x) 1-β , so that E |L| -1 β ≤ E sup x L 1 (x) 1-β β = 1 -β β +∞ 0 t 1 β -2 P sup x L 1 (x) ≥ t dt .
Therefore it suffices to prove that there exists a constant c > 0 such that

P sup x L 1 (x) ≥ t ≤ 2 exp(-ct) for all t > 0 . (5)
This follows from stronger results proved in [START_REF] Lacey | Large deviations for the maximum local time of stable Lévy processes[END_REF], but for sake of completeness, let us give a soft argument here. For a > 0, let τ a := inf {t : sup x L t (x) ≥ a}. The random variable τ a is a stopping time, and by continuity of t → sup x L t (x), sup x L τa (x) = a on {τ a < ∞}. It follows then from the inequality

sup x L t+s (x) ≤ sup x L t (x) + sup x (L t+s (x) -L t (x)) ,
and from the strong Markov property, that for any a > 0 and b > 0,

P sup x L 1 (x) ≥ a + b = P τ a ≤ 1 ; sup x L 1 (x) ≥ a + b ≤ E 1 {τa≤1} P Uτ a sup x L 1 (x) ≥ b ,
where for any v, P v denotes the law of the process U starting from v. By translation invariance, the law of sup x L 1 (x) does not depend on the starting point of U . Therefore, for any a > 0 and b > 0,

P sup x L 1 (x) ≥ a + b ≤ P [τ a ≤ 1] P sup x L 1 (x) ≥ b = P sup x L 1 (x) ≥ a P sup x L 1 (x) ≥ b .
(6) Let M > 0 be a median of sup x L 1 (x). By [START_REF] Borodin | The asymptotic behavior of local times of recurrent random walks with infinite variance[END_REF], for all t > 0,

P sup x L 1 (x) ≥ t ≤ P sup x L 1 (x) ≥ M ⌊t/M ⌋ ≤ 1 2 ⌊t/M ⌋ ,
which ends the proof of (5).

2.2.

A first reduction.

Lemma 5. Let n ≥ 1 and x ∈ Z be given.

• If P [nξ 1 -x / ∈ dZ] = 1, then P(Z n = x) = 0. • If P [nξ 1 -x ∈ dZ] = 1, then P(Z n = x) = d 2π π d -π d exp(-itx)E   y∈Z ϕ ξ (tN n (y))   dt .
Proof. We have

P(Z n = x) = 1 2π 2π 0 exp(-itx)ϕ n (t) dt ,
where ϕ n is the characteristic function of Z n given by

ϕ n (t) := E   y∈Z ϕ ξ (tN n (y))   for all t ∈ R.
Notice that e ] almost surely. Hence, for any integer m ≥ 0 and any u ∈ R,

ϕ ξ 2mπ d + u = ϕ ξ 2π d m ϕ ξ (u).
Therefore

P(Z n = x) = 1 2π d-1 k=0 π d -π d exp -i t + 2kπ d x ϕ n 2kπ d + t dt = 1 2π d-1 k=0 π d -π d exp(-itx) exp -i 2kπ d x E y ϕ ξ 2π d kNn(y) ϕ ξ (tN n (y)) dt = 1 2π d-1 k=0 exp -i 2kπ d x ϕ ξ 2π d kn π d -π d exp(-itx)ϕ n (t) dt, since y N n (y) = n. Moreover, e -i 2π d x ϕ ξ 2π d n d = e -i2πx e 2iπnξ 1 = 1, thus e -i 2π d x ϕ ξ 2π d n is a d th root of the unity. Hence d-1 k=0 e -i 2kπ d x ϕ ξ 2π d kn = d if ϕ ξ 2π d n e -i 2π d x = 1, 0 otherwise. Since ϕ ξ 2π d = e 2iπξ 1 d
a.s., the lemma follows. Lemma 6. For every n ≥ 1 and γ > 0, set

Ω n = Ω n (γ) := R n ≤ n 1 α +γ
and sup

y =z |N n (y) -N n (z)| |y -z| α-1 2 ≤ n (1-1 α +γ)/2
.

Then P(Ω n ) = 1 -o(n -δ ). Moreover, given η ≥ γ max(α/2, 2(β -1)/β), the following also holds on Ω n :

N * n ≤ n 1-1 α +η and V n := z N β n (z) ≥ n δβ-ηβ 2 if β > 1 n δβ-η(1-β) if β ≤ 1. ( 7
)
Proof. We prove in the appendix that for every γ > 0, there exists C > 0 such that

P (R n ≤ E[R n ]n γ ) = 1 -O(e -Cn γ ).
Since there exists c > 0 such that E[R n ] ∼ cn 1 α (see [START_REF] Spitzer | Principles of random walks[END_REF] p.36), we conclude that

P(R n ≤ n 1 α +γ ) = 1 -o(n -δ ).

Now let us prove that

P sup y =z |N n (y) -N n (z)| |y -z| α-1 2 ≥ n 1-1 α +γ = o(n -δ ).
According to the proof of Proposition 5.4 in [START_REF] Gall | The range of stable random walks[END_REF], we have :

E[|S n | p ] = O(n p α
), for all p ∈ (1, α). Then Doob's inequality gives that, for all δ ′ > δ/p,

P( sup k=1,...,n |S k | ≥ n 1 α +δ ′ ) = O(n -pδ ′ ) = o(n -δ ).

So we can restrict ourselves to the set A

n := {sup k=1,...,n |S k | < n 1 α +δ ′ }. But on A n , if N n (z) > 0 then necessarily z ∈ (-n 1 α +δ ′ , n 1 α +δ ′ ). Thus P sup y,z |N n (y) -N n (z)| |y -z| α-1 2 ≥ n 1-1 α +γ ; A n ≤ 5n 2 α +2δ ′ sup y =z P |N n (y) -N n (z)| |y -z| α-1 2 ≥ n 1-1 α +γ . (8)
Moreover the Markov inequality gives for all m ≥ 1:

P |N n (y) -N n (z)| |y -z| α-1 2 ≥ n 1-1 α +γ ≤ E[|N n (y) -N n (z)| 2m ] |y -z| (α-1)m n (1-1 α +γ)m
for all y = z.

In addition, according to [START_REF] Jain | Asymptotic behavior of the local time of a recurrent random walk[END_REF] (see the formula in the middle of page 77, with

m = O(n), a -1 m = O(n -1/α ) and Q(z) -1 = O(z α )), we have for all m ≥ 1, sup y =z E[|N n (y) -N n (z)| 2m ] |y -z| (α-1)m = O(n (1-1 α )m) ). ( 10
)
Thus if we take m > (δ + 2/α + 2δ ′ )/γ, then by using ( 8), ( 9) and ( 10), we get

P sup y =z |N n (y) -N n (z)| |y -z| α-1 2 ≥ n 1-1 α +γ = O n 2 α +2δ ′ n γm = o(n -δ ).
We now prove [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF], starting with the upper bound for N * n . For this let y 0 be such that N n (y 0 ) = N * n , and let z 0 be the closest point to y 0 such that

N n (z 0 ) = 0. Then on Ω n , |y 0 -z 0 | ≤ R n ≤ n 1 α +γ , and thus N n (y 0 ) ≤ |y 0 -z 0 | α-1 n 1-1 α +γ ≤ n ( 1 α +γ)(α-1) n 1-1 α +γ = n 1-1 α + αγ 2 . (11)
The desired upper bound for N * n follows if η ≥ αγ/2. To prove the lower bound for V n , we use the fact that n = y N n (y). When β > 1, this gives by using Hölder's inequality:

n ≤ z N β n (z) 1 β R 1-1 β n ≤ (V n ) 1 β n ( 1 α +γ) 1-1 β . Hence V 1 β n ≥ n δ-γ β-1
β , and the desired lower bound for

V n follows if 2(β -1)γ ≤ ηβ. When β ≤ 1, we write n = y N n (y) ≤ V n (N * n ) 1-β ,
and the desired lower bound follows from the upper bound for N * n proved just above.

2.4. Scheme of the proof. Let η > 0. Set γ := ηβ/2. We observe that γ ≤ η and that [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF] holds with this choice of (η, γ). We also set

η := η if β ≥ 1 η/β if β < 1.
By Lemmas 5 and 6, we have to estimate

d 2π π d -π d e -it⌊n δ x⌋ E y ϕ ξ (tN n (y))1 Ωn dt .
This is done in several steps presented in the following propositions.

Proposition 7. Let η ∈ 0, 1 2α(β+1) . Then, we have d 2π |t|≤n -δ+η e -it⌊n δ x⌋ E y ϕ ξ (tN n (y))1 Ωn dt = d C(x) n δ + o(n -δ ) , uniformly in x ∈ R.
Recall next that the characteristic function φ of the stable distribution S β has the following form :

φ(u) = e -|u| β (A 1 +iA 2 sgn(u)) , for some 0 < A 1 < ∞, |A -1 1 A 2 | ≤ | tan(πβ/2)|. It follows that the characteristic function ϕ ξ of ξ 1 satisfies: 1 -ϕ ξ (u) ∼ |u| β (A 1 + iA 2 sgn(u)) when u → 0. ( 12
)
Therefore there exist constants ε 0 > 0 and σ > 0 such that

max(|φ(u)|, |ϕ ξ (u)|) ≤ exp -σ|u| β for all u ∈ [-ε 0 , ε 0 ]. ( 13
)
Since ϕ ξ (t) = ϕ ξ (-t) for every t ≥ 0, the following propositions achieve the proof of Theorem 1:

Proposition 8. Let η be as in Proposition 7. Then there exists c > 0 such that

ε 0 n -1+ 1 α -η n -δ+η E y |ϕ ξ (tN n (y))|1 Ωn dt = o(e -n c ).
Proposition 9. Let η be as in Proposition 7 and let ε ∈ η, α-1 α(3+2β(α-1)) be given. Then there exists c > 0 such that

n -1+ 1 α +ε ε 0 n -1+ 1 α -η E y |ϕ ξ (tN n (y))|1 Ωn dt = o(e -n c ) . Proposition 10. Let η be such that γ < min 1 2α 2 , 1 2 α-1 α
and let ε ∈ ( 2α β + 1)γ, 1 -1 α be given. Then there exists c > 0 such that

π d n -1+ 1 α +ε E y |ϕ ξ (tN n (y))|1 Ωn dt = o(e -n c ).
To end the proof of Theorem 1, we observe that there exists (η, ε) satisfying all the hypotheses of these propositions (by taking η > 0 small enough and ε <

α-1 α(3+2β(α-1)) large enough). 2.5. Proof of Proposition 7. Remember that V n = z∈Z N β n (z).
We start by a preliminary lemma.

Lemma 11. If β > 1, then sup n E   n δ V 1 β n β β-1   < +∞. If β ≤ 1, then for all p ≥ 1, sup n E n δ V 1 β n p < +∞.
A direct consequence of this lemma is that the sequence (n δ V

-1 β n , n ≥ 1) is uniformly integrable.
Proof. We start with the case β > 1. We already observed in the proof of Lemma 6 that for every n ≥ 1,

n ≤ V 1 β n R 1-1 β n . But it is proved in [27] Equation (7.a) that E[R n ] = O(n 1 α
). The result follows. We suppose now that β ≤ 1. Since we have

n = x N n (x) ≤ V n (N * n ) 1-β , (14)
we get

n δ V 1/β n ≤ N * n n 1-1 α 1 β -1 . ( 15
)
We use next the fact that N * n is a subadditive functional:

N * n+m ≤ N * n + N * m • θ n , (16)
where

N * m • θ n := sup x m-1 k=0 1 {S n+k =x} = sup x m-1 k=0 1 {S n+k -Sn=x} , is independent of σ(S 0 , • • • , S n-1 ). Moreover, 0 ≤ N * n+1 -N * n ≤ 1.
Therefore, we can prove in exactly the same way as for the range (see (46) in the appendix), that

P (N * n ≥ a + b) ≤ P (N * n ≥ a) P (N * n ≥ b) for all a, b ∈ N . ( 17
)
Now it is known (see for example [START_REF] Borodin | The asymptotic behavior of local times of recurrent random walks with infinite variance[END_REF]) that

N * n /n 1-1/α converges in distribution toward sup x L 1 (x). Let t > 0, be such that P [sup x L 1 (x) ≥ t] ≤ 1/2. Since lim n→∞ P N * n ≥ tn 1-1/α ≤ P sup x L 1 (x) ≥ t ≤ 1/2,
we obtain that for n large enough, P N * n ≥ tn 1-1/α ≤ 2/3. Hence for n large enough, and all p ≥ 1,

E N * n n 1-1/α p = p ∞ 0 x p-1 P N * n ≥ xn 1-1/α dx ≤ pt p ∞ 0 u p-1 P N * n ≥ tn 1-1/α u du ≤ pt p ∞ 0 u p-1 P N * n ≥ tn 1-1/α ⌊u⌋ du ≤ pt p ∞ 0 u p-1 2 3 ⌊u⌋ du , (18)
where the first inequality in (18) comes from [START_REF] Feller | An introduction to probability theory and its applications[END_REF]. Thus, for all p ≥ 1,

sup n E N * n n 1-1/α p < ∞ . (19)
The lemma now follows from [START_REF] Dombry | A functional approach for random walks in random sceneries Electronic[END_REF].

The next step is the Lemma 12. Under the hypotheses of Proposition 7, we have

|t|≤n -δ+η e -it⌊n δ x⌋ E y ϕ ξ (tN n (y)) -e -|t| β Vn(A 1 +iA 2 sgn(t)) 1 Ωn dt = o(n -δ ) ,
uniformly in x ∈ R, where A 1 and A 2 are the constants appearing in [START_REF] Cohen | Convergence of dependent walks in a random scenery to fBm-local time fractionnal stable motions[END_REF].

Proof. It suffices to prove that

|t|≤n -δ+η E y ϕ ξ (tN n (y))1 Ωn -E e -|t| β Vn(A 1 +iA 2 sgn(t)) 1 Ωn dt = o(n -δ ) . Set E n (t) := y ϕ ξ (tN n (y)) - y exp -|t| β N β n (y)(A 1 + iA 2 sgn(t)) .
Observe that

E n (t) = y z<y ϕ ξ (tN n (z)) ϕ ξ (tN n (y)) -e -|t| β N β n (y)(A 1 +iA 2 sgn(t)) × z>y e -|t| β N β n (z)(A 1 +iA 2 sgn(t)) . But on Ω n , if |t| ≤ n -δ+η , then |t|N n (z) ≤ n η+η-1 αβ . ( 20
)
This implies in particular that |t|N n (z) < ε 0 for n large enough, since the hypothesis on η implies η + η < 1/(αβ). Thus by using ( 13) we get

|E n (t)| ≤ y ϕ ξ (tN n (y)) -exp -|t| β N β n (y)(A 1 + iA 2 sgn(t)) exp   -σ|t| β z =y N β n (z)   ,
for n large enough. Observe next that [START_REF] Cohen | Convergence of dependent walks in a random scenery to fBm-local time fractionnal stable motions[END_REF] implies

ϕ ξ (u) -exp -|u| β (A 1 + iA 2 sgn(u) ≤ |u| β h(|u|) for all u ∈ R,
with h a continuous and monotone function on [0, +∞) vanishing in 0. Therefore by using [START_REF] Guillotin-Plantard | Limit theorem for random walk in weakly dependent random scenery[END_REF] we get

|E n (t)| ≤ |t| β h(n η+η-1 αβ ) y N β n (y) exp   -σ|t| β z =y N β n (z)   .
Now on Ω n , according to [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF] and the hypothesis on η, if n is large enough,

z =y N β n (z) ≥ V n /2
for all y ∈ Z.

By using this and the change of variables

v = tV 1/β n , we get |t|≤n -δ+η E [|E n (t)|1 Ωn ] dt ≤ h(n η+η-1 αβ )E[V -1/β n ] R |v| β exp -σ|v| β /2 dv = o(E[V -1/β n ]),
which proves the result according to Lemma 11.

Finally Proposition 7 follows from the Lemma 13. Under the hypotheses of Proposition 7, we have

d 2π |t|≤n -δ+η e -it⌊n δ x⌋ E e -|t| β Vn(A 1 +iA 2 sgn(t)) 1 Ωn dt = d C(x) n δ + o(n -δ ) , uniformly in x ∈ R. Proof. Set I n,x := |t|≤n -δ+η
e -it⌊n δ x⌋ e -|t| β Vn(A 1 +iA 2 sgn(t)) dt.

Since | n δ x -n δ x| ≤ 1, for all n and x, it is immediate that

I n,x = |t|≤n -δ+η e -itn δ x e -|t| β Vn(A 1 +iA 2 sgn(t)) dt + O(n -2δ+2η ).
But 2η < 1/(αβ) < δ by hypothesis. So actually

I n,x = |t|≤n -δ+η e -itn δ x e -|t| β Vn(A 1 +iA 2 sgn(t)) dt + o(n -δ ).
Next, after some changes of variables, we get:

|t|≤n -δ+η e -itn δ x e -|t| β Vn(A 1 +iA 2 sgn(t)) dt = n -δ 2π n δ V 1/β n f β n δ x V 1/β n -J n,x , (21)
where

J n,x := |v|≥n η e -ivx e -|v| β Vn n βδ (A 1 +iA 2 sgn(v)) dv. Now it is known that W n := n δ V -1/β n converges in distribution, as n → ∞, toward W := |L| -1 β
(see [START_REF] Chen | Large deviations for local times of stable processes and stable random walks in 1 dimension Electron[END_REF] Lemma 14 or [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] Lemma 6). Then by Skorohod's representation Theorem, we can find a sequence ( W n , n ≥ 1) and W distributed respectively as (W n , n ≥ 1) and W such that W n converges almost surely toward W . Moreover, Lemma 11 ensures that the sequence ( W n , n ≥ 1) is uniformly integrable, so actually the convergence holds in L 1 . Let us deduce that

E[g x (W n )] = E[g x (W )] + o(1), (22)
where g x : z → zf β (xz) and the o(1) is uniform in x. First

|E[g x (W n )] -E[g x (W )]| ≤ sup x,z∈R |(g x ) ′ (z)|E[| W n -W |] ≤ sup u |f β (u) + uf ′ β (u)|E[| W n -W |].
But remember that

f β (u) = 1 2π R e itu e -|t| β (A 1 +iA 2 sgn(t)) dt .
So after differentiation under the integral sign and integration by parts we get

uf ′ β (u) = - 1 2π R e itu (1 -βsgn(t)|t| β (A 1 + iA 2 sgn(t)))e -|t| β (A 1 +iA 2 sgn(t)) dt.
In particular sup u |f β (u) + uf ′ β (u)| is finite, and this proves [START_REF] Jain | Asymptotic behavior of the local time of a recurrent random walk[END_REF]. In view of [START_REF] Hall | Martingale limit theory and its application[END_REF] it only remains to prove that E[J n,x 1 Ωn ] = o(1). But this follows from the basic inequality

E[|J n,x 1 Ωn |] ≤ |v|≥n η E e -A 1 |v| β Vn n βδ 1 Ωn dv,
and from the lower bound for V n given in [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF].

2.6. Proof of Proposition 8. Recall that on Ω n , N n (y) ≤ n 1-1 α +η , for all y ∈ Z. Hence by [START_REF] Csáki | Strong invariance for local times Z. Wahrscheinlichkeitstheorie verw[END_REF],

ε 0 n -1+ 1 α -η n -δ+η E y |ϕ ξ (tN n (y))|1 Ωn dt ≤ ε 0 n -1+ 1 α -η n -δ+η E exp -σt β V n 1 Ωn dt .
But on Ω n , we can also use the lower bound for V n given in [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF], which implies that

ε 0 n -1+ 1 α -η n -δ+η E y |ϕ ξ (tN n (y))|1 Ωn dt ≤ e -σn cη ,
for some constant c > 0, depending on β. This proves the proposition. 2.7. Proof of Proposition 9. First note that by using again [START_REF] Csáki | Strong invariance for local times Z. Wahrscheinlichkeitstheorie verw[END_REF] we get

y |ϕ ξ (tN n (y))| ≤ exp    -σt β z:Nn(z)≤ε 0 n 1-1 α -ε N β n (z)    for all t ≤ n -1+ 1 α +ε . (23)
The proof will then be a consequence of the Lemma 14. Under the hypotheses of Proposition 9, for n large enough and on Ω n , we have

# z : ε 0 10 n 1-1 α -ε ≤ N n (z) ≤ ε 0 n 1-1 α -ε ≥ ε 0 10 2 α-1 n 1 α -2ε+γ α-1 .
Indeed according to this lemma and ( 23), we get for n large enough and on Ω n ,

y |ϕ ξ (tN n (y))| ≤ exp -σ ′ n -β(1-1 α +η) n 1 α -2ε+γ α-1 n β(1-1 α -ε) ≤ exp -σ ′ n 1 α -β(η+ε)-2ε+γ α-1 for all ε 0 n -1+ 1 α -η ≤ t ≤ n -1+ 1 α +ε ,
for some constant σ ′ > 0. This proves Proposition 9, since the hypothesis on ε and γ implies that 1

α -β(η + ε) - 2ε + γ α -1 > 1 α -2βε - 3ε α -1 > 0.
Proof of Lemma 14. Let y 1 be such that

N n (y 1 ) = N * n = sup z N n (z). Since n = z N n (z) ≤ N * n R n , we have N n (y 1 ) ≥ n 1-1 α -γ , on Ω n . Set y 0 := min y ≥ y 1 : N n (y) ≤ ε 0 2 n 1-1 α -ε .
Observe that y 0 > y 1 for n large enough, since ε > γ by hypothesis. In particular

N n (y 0 -1) > ε 0 2 n 1-1 α -ε ≥ N n (y 0 ). But on Ω n , N n (y 0 -1) -N n (y 0 ) ≤ n (1-1 α +γ)/2 .
Moreover, the hypotheses made on γ and ε imply that γ < (1

-1/α)/3 and ε < (1 -1/α)/3. Thus ε < (1 -1/α -γ)/2, or equivalently (1 -1/α + γ)/2 < 1 -1/α -ε. Therefore ε 0 4 n 1-1 α -ε ≤ N n (y 0 ) ≤ ε 0 2 n 1-1 α -ε , (24)
for n large enough. Next if

|y 0 -z| ≤ ε 0 10 2 α-1 n 1 α -2ε+γ α-1 , then on Ω n , |N n (z) -N n (y 0 )| ≤ |y 0 -z| α-1 n 1-1 α +γ ≤ ε 0 10 n 1-1 α -ε .
Together with [START_REF] Khoshnevisan | A law of the iterated logarithm for stable processes in random scenery[END_REF], this proves the lemma.

2.8. Proof of Proposition 10. Let M and N be two positive integers such that P(X 1 = N ) > 0 and P(X 1 = -M ) > 0. We denote by C + the (M + N )-uple (N, ..., N, -M, ..., -M ) in which N is repeated M times and then -M is repeated N times. We denote by C -the "symmetric" (M + N )-uple (-M, ..., -M, N, ..., N ) in which -M is repeated N times and then N is repeated M times. Set T := M + N and observe that

p := P((X 1 , ..., X T ) = C + ) = P((X 1 , ..., X T ) = C -) > 0.
Let us notice that (X 1 , ..., X T ) = C + corresponds to a trajectory going up to M N (in M steps) and then coming back down to 0 (in N steps). Analogously, (X 1 , ..., X T ) = C -corresponds to a trajectory that goes down to -M N (in N steps) and comes back up to 0 (in M steps).

We introduce now the event

D n := C n > np 2T ,
where

C n := # k = 0, ..., n T -1 : (X kT +1 , . . . , X (k+1)T ) = C ± .
Since the sequences (X kT +1 , . . . , X (k+1)T ), for k ≥ 0, are independent of each other, Chernoff's inequality implies that there exists c > 0 such that

P(D n ) = 1 -o(e -cn ).
We introduce now the notion of "peak". We say that there is a peak based on y at time n if S n = y and (X n+1 , . . . , X n+T ) = C ± . We will see (in Lemma 15 below) that, on Ω n ∩ D n , there is a large number of y ∈ Z on which are based a large number of peaks. For any y ∈ Z, let Proof. Note that C n (y) ≤ N n (y) for all y ∈ Z.

Thus on Ω n ∩ D n , np 2T ≤ y∈Z : Cn(y)<n 1-1 α -2γ C n (y) + y∈Z : Cn(y)≥n 1-1 α -2γ C n (y) ≤ n 1-1 α -2γ R n + N * n p n ≤ n 1-γ + p n n 1-1 α + αγ 2 ,
according to [START_REF] Chen | Large deviations for local times of stable processes and stable random walks in 1 dimension Electron[END_REF]. This proves the lemma.

We have proved that, if n is large enough, the event Ω n ∩ D n is contained in the event

E n := {p n ≥ 3N M n 1 α -αγ }. Now, on E n , we define Y i for i = 1, . . . , n 1 α -αγ , by Y 1 := min y ∈ Z : C n (y) ≥ n 1-1 α -2γ , and
Y i+1 := min y ≥ Y i + 3N M : C n (y) ≥ n 1-1 α -2γ for i ≥ 1.
The Y i 's are sites on which at least n 1-1 α -2γ peaks are based and are such that

|Y i -Y j | ≥ 3N M , if i = j. For every i = 1, . . . , n 1 α -αγ , let t 1 i , . . . , t n 1-1 α -2γ
i be the n 1-1 α -2γ first times (which are multiples of T ) when a peak is based on the site Y i . We also define N 0 n (Y i + N M ) as the number of visits of S before time n to Y i + N M , which do not occur during the time intervals [t j i , t j i + T ], for j ≤ n 1-1 α -2γ .

Lemma 16. Conditionally to the event

E n , ((N n (Y i + M N )-N 0 n (Y i + M N ), i ≥ 1) is a sequence of independent identically distributed random variables with binomial distribution B n 1-1 α -2γ ; 1 2 . Moreover this sequence is independent of ((N 0 n (Y i + M N ), i ≥ 1).
Proof. On E n , we have

N n (Y i + M N ) -N 0 n (Y i + M N ) = ⌊n 1-1 α -2γ ⌋ j=1 1 {(X t j i +1 ,...,X t j i +T )∈C + } ,
since the peaks based on the other Y k 's cannot pass through

Y i + M N . But conditionally to E n , the sequence 1 {(X t j i +1 ,...,X t j i +T )∈C + } i,j
is a sequence of independent Bernoulli random variables with parameter 1/2, which is independent of (X k , k ∈ i,j [t j i , ..., t j i + T ]). Since N 0 n (Y i + M N ) only depends on the values of the X k 's for k ∈ i,j [t j i , ..., t j i + T ], the result follows.

Let now ρ := sup{|ϕ ξ (u)| : d u, 2π d Z ≥ ε 0 }. According to Formula (13),

|ϕ ξ (u)| ≤ ρ1 {d(u, 2π d Z)≥ǫ 0 } + exp -σd u, 2π d Z β 1 {d(u, 2π d Z)<ǫ 0 } ≤ exp -σn -1 α +2αγ , as soon as d u, 2π d Z ≥ n -1 αβ + 2αγ
β , and ρ ≤ exp -σn -1 α +2αγ . But recall that ρ < 1 and 2α 2 γ < 1. Therefore, for n large enough,

z |ϕ ξ (tN n (z))| ≤ exp -σn -1 α +2αγ # z : d tN n (z), 2π d Z ≥ n -1 αβ + 2αγ β . (25)
Then notice that

d tN n (z), 2πZ d ≥ n -1 αβ + 2αγ β ⇐⇒ N n (z) ∈ I := k∈Z I k , (26)
where for all k ∈ Z,

I k := 2kπ dt + n -1 αβ + 2αγ β t , 2(k + 1)π dt - n -1 αβ + 2αγ β t .
In particular R \ I = k∈Z J k , where for all k ∈ Z,

J k := 2kπ dt - n -1 αβ + 2αγ β t , 2kπ dt + n -1 αβ + 2αγ β t .
Lemma 17. Under the hypotheses of Proposition 10, for every i ≤ n 1 α -αγ , t ∈ (n -1+ 1 α +ε , π/d) and n large enough,

P N n (Y i + M N ) ∈ I | E n , N 0 n (Y i + M N ) ≥ 1 3 almost surely.
Assume for a moment that this lemma holds true and let us finish now the proof of Proposition 10. Lemmas 16 and 17 ensure that conditionally to E n and ((N 0 n (Y i + M N ), i ≥ 1), the events {N n (Y i + M N ) ∈ I}, i ≥ 1, are independent of each other, and all happen with probability at least 1/3. Therefore, since Ω n ∩ D n ⊆ E n , there exists c > 0, such that

P Ω n ∩ D n , #{i : N n (Y i + M N ) ∈ I} ≤ n 1 α -αγ 4 ≤ P B n ≤ n 1 α -αγ 4 = o(exp(-cn)),
where for all n ≥ 1, B n has binomial distribution B n

1 α -αγ ; 1 3 . But if #{z : N n (z) ∈ I} ≥ n 1 α -αγ /4
, then by ( 25) and ( 26) there exists a constant c > 0, such that

z |ϕ ξ (tN n (z))| ≤ exp -cn 1 α -αγ n -1 α +2αγ ,
which proves Proposition 10.

Proof of Lemma 17. First notice that by Lemma 16, for any H ≥ 0,

P(N n (Y i + M N ) ∈ I | E n , N 0 n (Y i + M N ) = H) = P (H + b n ∈ I) , (27)
where b n is a random variable with binomial distribution B n 1-1 α -2γ ; 1 2 . We will use the following result whose proof is postponed. Lemma 18. Under the hypotheses of Proposition 10, for every t ∈ (n -1+ 1 α +ε , π/d) and for n large enough, the following holds:

(i) For any integer k such that all the elements of I k -H are smaller than 1 2 n 1-1 α -2γ ,

P(b n ∈ (I k -H)) ≥ P(b n ∈ (J k -H)).
(ii) For any integer k such that all the elements of I k -H are larger than 1 2 n 1-1 α -2γ ,

P(b n ∈ (I k -H)) ≥ P(b n ∈ (J k+1 -H)).
Now call k 0 the largest integer satisfying the condition appearing in (i) and k 1 the smallest integer satisfying the condition appearing in (ii). We have

k 1 = k 0 + 1 or k 1 = k 0 + 2.
According to Lemma 18, we have

P (H + b n ∈ I) ≥ k≤k 0 P (H + b n ∈ I k ) + k≥k 1 P (H + b n ∈ I k ) ≥ k≤k 0 P (H + b n ∈ J k ) + k≥k 1 P (H + b n ∈ J k+1 ) = P(H + b n ∈ I) -P(H + b n ∈ J k 0 +1 ∪ J k 1 ).
Hence,

P (H + b n ∈ I) ≥ 1 2 [1 -P(H + b n ∈ J k 0 +1 ∪ J k 1 )] . Let bn := 2 b n -1 2 n 1-1 α -2γ n 1-1 α -2γ -1/2
, so that bn converges in distribution to a standard normal variable, whose distribution function is denoted by Φ. The interval J k 1 being of length 2n

-1 αβ + 2αγ β /t, P(H + b n ∈ J k 1 ) = P( bn ∈ [m n , M n ]) , with M n -m n = 4 n -1 αβ + 2αγ β t n 1-1 α -2γ ≤ Φ(M n ) -Φ(m n ) + C n 1-1 α -2γ
(by the Berry-Esseen inequality)

≤ M n -m n √ 2π + C n 1-1 α -2γ ≤ C ′ n 1 2 + 1 2α +γ+ 2αγ β -1 αβ -1 α -ε + C n 1-1 α -2γ
, for t ≥ n -1+ 1 α +ε , and some constants C > 0 and

C ′ > 0. Since α ≤ 2, β ≤ 2, γ < 1 2 α-1
α , and ε > 2αγ/β + γ by hypothesis, we conclude that P(H + b n ∈ J k 1 ) = o(1). The same holds for P(H + b n ∈ J k 0 +1 ), so that for n large enough,

P (H + b n ∈ I) ≥ 1 2 [1 -o(1)] ≥ 1 3 .
Together with [START_REF] Gall | The range of stable random walks[END_REF], this concludes the proof of Lemma 17.

Proof of Lemma 18. We only prove (i), since (ii) is similar. So let k be an integer such that all the elements of I k -H are smaller than 1 2 n 1-1 α -2γ . Assume that (J k -H)∩ Z contains at least one nonnegative integer (otherwise P(b n ∈ (J k -H)) = 0 and there is nothing to prove). Let z k denote the greatest integer in J k -H, so that by our assumption P(b n = z k ) > 0 (remind that 0 ≤ z k < 1 2 n 1-1 α -2γ ). By monotonicity of the function z → P(b n = z), for z ≤ 1 2 n 1-1 α -2γ , we get

P(b n ∈ J k -H) ≤ P(b n = z k )#((J k -H) ∩ Z) ≤ P(b n = z k ) 2n -1 αβ + 2αγ β t .
In the same way,

P(b n ∈ I k -H) ≥ P(b n = z k )#((I k -H) ∩ Z) ≥ P(b n = z k ) 2π dt - 2n -1 αβ + 2αγ β t .
Hence

P(b n ∈ I k -H) ≥ 2π dt -2n -1 αβ + 2αγ β t 2n -1 αβ + 2αγ β t P(b n ∈ J k -H) .
But π/(dt) ≥ 1 and 2α 2 γ < 1 by hypothesis. It follows immediately that for n large enough, we have 2n -1 αβ + 2αγ β < π/(2d), and so

2π dt - 2n -1 αβ + 2αγ β t ≥ 3π 2dt ≥ 1 + π 2dt ≥ π 2dt ≥ 2n -1 αβ + 2αγ β t .
This concludes the proof of the lemma.

Lattice case, α < 1: Proof of Theorem 2

We only sketch the proof, since it is very similar and simpler than in the case α > 1. In particular we keep the same notation, for instance for N * n , R n , V n , ε 0 ,... We first introduce the analogue Ω ′ n of Ω n :

Ω ′ n = Ω ′ n (ε) := {N * n ≤ n ε } , which is well defined for any ε. Note that on Ω ′ n , we have V n ≥ R n ≥ n 1-ε . ( 28
)
Since

N * n = sup n-1 k=0 [N n (S k ) -N k (S k )]
, we obtain that

P (N * n ≥ n ε ) ≤ nP sup m N m (0) ≥ n ε ≤ np n ε -1 0 ,
where p 0 := P (∃k ≥ 1 : S k = 0). Since α < 1, the random walk S is transient and

p 0 < 1. It follows that P(Ω ′ n ) = 1 -o(exp(-n c
)), for some constant c > 0, and we can restrict our study to this set. Moreover, it is known (see for instance the introduction in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] for an argument) that

1 n V n = 1 n y∈Z N β n (y) -→ n→+∞ E[ N β-1 ∞ (0)] = r -β a.s.. We claim now that n 1/β V -1/β n
, n ≥ 1 is uniformly integrable. Indeed, if β ≥ 1, this comes from the fact that V n is larger than n, and when β < 1, this follows from the Lemma 19. If β < 1, there exists γ > 0 such that

sup n E exp γ n V n < ∞ . ( 29
)
Proof. Since n = x N n (x), Hölder's inequality gives

n V n ≤ x N n (x) 2 n . Since 1 n x N n (x) 2 = 1 n n-1 k=0 N n (S k ) , Jensen's inequality gives exp γ x N n (x) 2 n ≤ 1 n n-1 k=0 exp (γN n (S k )) .
Hence,

E exp γ x N n (x) 2 n ≤ 1 n n-1 k=0 E [exp (γN n (S k ))] ≤ E exp γ N ∞ (0) .
Then, [START_REF] Matheron | Is transport in porous media always diffusive? A counterxample[END_REF] directly follows from the fact that N ∞ (0) is equal to 1 plus the sum of two independent geometric variables with positive parameter, and thus has finite exponential moments.

Let ε > 0 and η > 0 be such that η + ε < 1/β and ε < ηβ < 1/2. As in the proof of Proposition 7, we deduce that

d 2π |t|≤n -1 β +η e -it n 1 β x E y ϕ ξ (tN n (y)) dt = D(x) n 1 β + o(n -1 β ) ,
where the o(n -1/β ) is uniform in x. It remains to prove that

d 2π π d n -1 β +η E y ϕ ξ (tN n (y))1 Ω ′ n dt = o(n -1 β ). ( 30
)
As in the proof of Proposition 10 (see the beginning of Section 2.8. for the definitions of D n , C n , C n (y), . . .), we are led to prove that

d 2π π d n -1 β +η E y ϕ ξ (tN n (y))1 Ω ′ n ∩Dn dt = o(n -1 β ).
Let p ′ n := #{y : C n (y) ≥ 1} be the random variable equal to the number of sites of Z on which at least one peak is based. Let us notice that on Ω

′ n ∩ D n , we have np 2T ≤ C n = y C n (y) ≤ y : Cn(y)≥1 N n (y) ≤ p ′ n n ε . Thus Ω ′ n ∩ D n ⊆ E ′ n , where E ′ n := {p ′ n ≥ c 0 n 1-ε }, for a well chosen constant c 0 > 0. As in the proof of Proposition 10, we construct (Y i ) i such that C n (Y i ) ≥ 1 and Y i+1 -Y i > M N .
For every i, we define N 0 n (Y i + M N ) as the number of visits to the site Y i + M N without taking into account the possible visit during the first peak based on Y i . Next we see that, on

E ′ n , (N n (Y i + M N ) -N 0 n (Y i + M N ), i ≤ c 0 n 1-ε
) is a sequence of i.i.d. random variables with Bernoulli distribution with parameter 1/2. Let t ∈ n -1 β +η , π d . We define the good and bad intervals respectively by

I ′ k := 2kπ dt + 1 2 , 2(k + 1)π dt - 1 2 and J ′ k := 2kπ dt - 1 2 , 2kπ dt + 1 2 .
Set also I ′ := k∈Z I ′ k . We observe that J ′ k is an open interval of length 1 and

I ′ k is a closed interval of length 2π/(dt) -1 ≥ 1 (since t ≤ π/d). Hence, if N 0 n (Y i + M N ) is not in I ′ , then N 0 n (Y i + M N ) + 1 is in I ′ .
This ensures that, on E ′ n , N n (Y i + M N ) belongs to I ′ with probability at least 1/2. Therefore, as after Lemma 17, we get

P Ω n ∩ D n ; #{i : N n (Y i + M N ) ∈ I ′ } < c 0 n 1-ε 3 = o(n -1 β ).
Hence, we just have to prove that

d 2π π d n -1 β +η E y ϕ ξ (tN n (y))1 Hn,t dt = o(n -1 β ), with H n,t := #{y : N n (y) ∈ I ′ } ≥ c 0 n 1-ε

3

. As after Lemma 16, we notice that, if n is large enough, we have

d u, 2π d Z ≥ n -1 β +η 2 ⇒ |ϕ ξ (u)| ≤ exp - σ 2 β n -1+βη .
We notice also that if N n (y) ∈ I ′ , then d tN n (y), 2π d Z ≥ t/2, and thus d tN n (y), 2π d Z ≥ n -1 β +η /2. Now, on H n,t , we know that at least c 0 n 1-ε /3 sites y satisfy this property. Therefore

E y ϕ ξ (tN n (y))1 Hn,t ≤ exp - c 0 σ 2 β 3 n 1-ε n -1+βη = o(n -1 β ),
since ε < βη. This gives [START_REF] Pène | Transient random walk in Z 2 with stationary orientations[END_REF] and achieves the proof of Theorem 2.

4. The strongly nonlattice case: Proof of Theorem 3

We assume here that ξ is strongly nonlattice. In that case, there exist ε 0 > 0, σ > 0 and ρ < 1 such that

• |ϕ ξ (u)| ≤ ρ if |u| ≥ ε 0 , • |ϕ ξ (u)| ≤ exp(-σ|u| β ) if |u| < ε 0 .
Case α > 1. We use here the notations of Section 2 with the hypotheses on γ, η, η and ε of propositions 7, 8, 9 and 10. Let h 0 be the density of Polya's distribution:

h 0 (y) = 1 π 1 -cos(y) y 2 .
Its Fourier transform is ĥ0 (t) = (1 -|t|) + . For θ ∈ R, let h θ (y) = exp(iθy)h 0 (y) with Fourier transform ĥθ (t) = ĥ0 (t + θ). As is proved in [START_REF] Durrett | Probability: theory and examples[END_REF] (see the proof of Theorem 5.4 p.114), it is enough to show that for all θ ∈ R,

lim n→∞ n δ E h θ (Z n -n δ x) = C(x) ĥθ (0) . (31)
By Fourier inverse transform,

n δ E h θ (Z n -n δ x) = n δ 2π R e -iun δ x E x∈Z ϕ ξ (uN n (x)) ĥθ (u) du .
Since ĥθ ∈ L 1 , we can restrict our study to the event Ω n of Lemma 6. The part of the integral corresponding to |u| ≤ n -δ+η is treated exactly as in Proposition 7. The only change is that we have to check that

lim n→∞ n δ |u|≤n -δ+ η E e -|u| β Vn(A 1 +iA 2 sgn(u)) 1 Ωn ( ĥθ (u) -ĥθ (0)) du = 0 ,
which is obviously the case since 2η < δ, using the fact that ĥθ is a Lipschitz function. Now since ĥθ is bounded, the part corresponding to n -δ+η ≤ |u| ≤ n -1+ 1 α +ε doesn't need any additional treatment. Actually, the proofs of Propositions 8 and 9 only use the behavior of ϕ ξ around 0, which is the same in the lattice or nonlattice case.

We now turn our attention to the part of the integral corresponding to |u| ≥ n -1+ 1 α +ε and prove that lim

n→∞ n δ |u|≥n -1+ 1 α +ε e -iun δ x E x ϕ ξ (uN n (x))1 Ωn ĥθ (u) du = 0 . (32)
To this end, note that

E x ϕ ξ (uN n (x))1 Ωn ≤ E ρ #{x : |uNn(x)|≥ε 0 } 1 Ωn , and that on Ω n , for |u| ≥ n -1+ 1 α +ε , n = x N n (x) ≤ ε 0 |u| R n + N * n # {x : |uN n (x)| ≥ ε 0 } ≤ ε 0 n 1-ε+ηβ/2 + n 1-1 α +η # {x : |uN n (x)| ≥ ε 0 } .
Hence, since ε > ηβ/2, for n large enough, on Ω n , and for |u| ≥ n -1+ 1 α +ε ,

# {x : |uN n (x)| ≥ ε 0 } ≥ 1 2 n 1 α -η .
Therefore, for n large enough,

n δ |u|≥n -1+ 1 α +ε e -iun δ x E x ϕ ξ (uN n (x))1 Ωn ĥθ (u) du ≤ n δ ρ 1 2 n 1 α -η R ĥθ (u) du ,
which tends to zero since η < 1/α.

Case α < 1. Using the notations and hypotheses on ε, η, γ of Section 3, one has to prove that for all θ ∈ R and all x ∈ R,

lim n→∞ n 1/β R e -iun 1/β x E x ϕ ξ (uN n (x))1 Ω ′ n ĥθ (u) du = D(x) ĥθ (0) . (33)
Again, the only change in the proof concerns the part of the integral corresponding to |u| ≥ n -1/β+η . We use here the bound

|ϕ ξ (uN n (x))| ≤ exp(-σ|u| β N β n (x)) 1 {|uNn(x)|≤ε 0 } + ρ 1 {|uNn(x)|≥ε 0 } ≤ exp(-σn -1+ηβ N β n (x)) 1 {|uNn(x)|≤ε 0 } + ρ 1 {|uNn(x)|≥ε 0 } . If η < 1/β, then for n large enough, ρ ≤ exp(-σn -1+ηβ
). Therefore, if n is large enough, then for all x and u such that N n (x) ≥ 1 and |u| ≥ n -1/β+η , we have

|ϕ ξ (uN n (x))| ≤ exp(-σn -1+ηβ ) .
Hence,

E x ϕ ξ (uN n (x)) 1 Ω ′ n ≤ E exp(-σn -1+ηβ R n )1 Ω ′ n ≤ exp(-σn ηβ-ε ) . Therefore, since ε < ηβ, lim n→∞ n 1/β |u|≥n -1/β+η e -iun 1/β x E x ϕ ξ (uN n (x)) 1 Ω ′ n ĥθ (u) du = 0 .
This concludes the proof of Theorem 3.

5.

Random walks on randomly oriented lattices 5.1. Model and result. We consider parallel moving pavements with different fixed speeds, independently chosen at the beginning with the same distribution. We study the random walk (M n , n ≥ 0) representing the position of a walker who at each time stays on the same moving pavement with probability p ∈ (0, 1), or jumps to another one with probability 1 -p.

Let us define (M n , n ≥ 0) more precisely. Let µ X be a distribution on Z in the normal domain of attraction of a centered stable distribution with index 1 < α ≤ 2 and density function denoted by f α (•). Let also ξ := (ξ y , y ∈ Z) be a sequence of independent centered Z-valued random variables with distribution µ ξ belonging to the normal domain of attraction of a stable distribution with index 1 < β ≤ 2 and density function denoted by f β (•). For each y ∈ Z, ξ y will be the only horizontal displacement allowed on line y. Let p ∈ (0, 1). Given ξ, the random walk (M n = (M

(1) n , M (2)
n ), n ≥ 0) is a Markov chain starting from M 0 := (0, 0) and such that at time n + 1, it moves either horizontally of ξ M (2) n (with probability p) or makes a vertical jump with distribution µ X (with probability (1 -p)), i.e.

P M n+1 -M n = (ξ M (2) n , 0) | ξ, M 1 , ..., M n = p if ξ M (2) n = 0, P (M n+1 -M n = (0, x) | ξ, M 1 , ..., M n ) = (1 -p)µ X (x) if x = 0 and P (M n+1 = M n | ξ, M 1 , ..., M n ) = p + (1 -p)µ X (0) if ξ M (2) n = 0.
These random walks were first introduced by Campanino and Pétritis in [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF] in the particular case when p = 1/3 and when µ X and µ ξ are Rademacher distributions, i.e. take values ±1 with probability 1/2. They proved the transience of M as well as a law of large numbers. In [START_REF] Guillotin-Plantard | A functional limit theorem for a 2d-random walk with dependent marginals[END_REF], Guillotin-Plantard and Le Ny established the link between the Campanino and Pétritis random walk and the random walk in random scenery and proved a functional limit theorem for the first one. It was also conjectured there that the probability of return to the origin of the Campanino and Pétritis random walk is equivalent to a constant times n -5/4 . We prove this result here, as well as a generalization to the case of the random walks M considered above.

To state our result, we will use the following representation of M :

Let X := (X n , n ≥ 1) be a sequence of independent random variables with distribution µ X . The random variable X n corresponds to the vertical move at time n which will be chosen with probability 1 -p. Let also (ε n , n ≥ 0) be a sequence of independent Bernoulli random variables with parameter p, i.e. such that P(ε 1 = 1) = 1 -P(ε 1 = 0) = p, and independent of X. If ε n = 1, the particle M moves horizontally at time n, otherwise it moves vertically. We then first define S by S 0 := 0 and

S n := n k=1 Y k for n ≥ 1 ,
where Y k := X k (1 -ε k ). We next define Z by Z 0 := 0 and

Z n := n k=1 ξ S k-1 ε k = y∈Z ξ y N n (y) for n ≥ 1 ,
where N n (y) := #{k = 1, ..., n : S k-1 = y and ε k = 1} .

Then it is straightforward that the couple ( Z, S) has the same distribution as M .

We just notice that the process S in this section is not exactly the same as in the previous sections (it is the same if we replace X by Y ). However, we use the same notation just for convenience.

Now it is known that (n -1/α S [nt] , t ≥ 0) converges in distribution, when n → ∞, to a Lévy process Ũ = ( U t , t ≥ 0) where U = (1-p)

1
α U and U is the process introduced in the introduction. We will use the fact that (n -1/α S [nt] , t ≥ 0 | S n = 0) converges in distribution to U 0 = ( U 0 t , t ≥ 0) the associated bridge, i.e. the process U starting from 0 and conditioned by U 1 = 0. Let (L 0 t (x), t ∈ [0, 1], x ∈ R) be the local time process of U 0 and set

|L 0 | β := R (L 0 1 (x)) β dx 1/β .
Let ϕ ξ be the characteristic functions of ξ 1 . Recall that d is the positive integer such that {u : |ϕ ξ (u)| = 1} = (2π/d)Z. Let d 0 be the smallest positive integer m such that ϕ ξ (2π/d) m = 1 and let d 1 be the greatest common divisor of {m ≥ 1 : P(X 1 + ... + X m ) > 0}.

Theorem 20. Assume that d 1 is a multiple of d 0 , and let

E = dp -1 f α (0)f β (0)E(|L 0 | -1 β ). Then, P(M n = (0, 0)) = E × n -1-1 αβ + o(n -1-1 αβ ) if n is a multiple of d 0 ; 0 otherwise.
Remark 21. In the case of the Campanino and Pétritis random walk, d 0 = d 1 = 2. So the hypothesis of the theorem is satisfied.

Remark 22. A corollary of our result is that the processes M considered here are transient, this can be seen by using Borel-Cantelli lemma.

Remark 23. It is most likely that an analogue result can be proved when α < 1 or β ≤ 1. We leave the details to the interested reader. In the same way one could certainly obtain similar estimates for the probabilities of return in ([n δ x], [n 1/α y]), with a constant E depending on x and y. For γ > 0, set Ω n = Ω n (γ) := A n ∩ B n ∩ C n , where

A n := R n ≤ n 1 α +γ and sup y N n (y) ≤ n 1-1 α +γ , B n := n k=1 ε k ≥ np 2 ,
and

C n := sup y =z | N n (y) -N n (z)| |y -z| α-1 2 ≤ n (1-1 α +γ)/2 .
Lemma 24. For all γ > 0, P(

Ω n ) = 1 -o(n -1-1 αβ ).
Proof. According to the proof of Lemma 6, P(R n ≤ n

1 α +γ ) = 1-o(n -1-1 αβ
). Moreover, according to the proof of Lemma 11 (see ( 19)), we have for all ν ≥ 1,

E sup y N ν n (y) = O n ν(1-1 α ) . (34)
Hence by the use of the Markov inequality, we get

P sup y∈Z N n (y) ≥ n 1-1 α +γ = o(n -1-1 αβ ). It follows that P(A n ) = 1 -o(n -1-1 αβ ).
Next it is well known that P(

B n ) = 1 -o(n -1-1 αβ ).
Finally, as in the proof of Lemma 6, the estimate of P(C n ) comes from the following lemma:

Lemma 25. For any integer ν ≥ 1, there exists a constant C ν > 0 such that, for every n ≥ 1 and every x, y ∈ Z

E ( N n (x) -N n (y)) 2ν ≤ C ν |x -y| ν(α-1) n ν(1-1 α ) .
Proof. Let x and y be fixed, with x = y (otherwise, there is nothing to prove). We have

N n (x) = pN n (x) + n k=1 1 {S k-1 =x} ǭk , (35)
where ǭk = 1 {ǫ k =1} -p. Set H n (x) := n k=1 1 {S k-1 =x} ǭk . For all x ∈ Z, (H n (x), n ≥ 1) is a martingale with respect to the filtration F n = σ(X k , ǫ k , k ≤ n). Hence, (H n (x) -H n (y), n ≥ 1) is a martingale as well. According to the Bürkholder's inequality (see [START_REF] Hall | Martingale limit theory and its application[END_REF] Theorem 2.11. p.23), for all integer ν ≥ 1, there exists a constant C = C(ν) such that for all n ≥ 1,

E (H n (x) -H n (y)) 2ν 1 2ν ≤ C    E n k=1 E(d 2 k (x, y) | F k-1 ) ν 1 2ν + E sup k=1,...,n |d k (x, y)| 2ν 1 2ν    ,
where d k (x, y) is the martingale increment

d k (x, y) := H k (x) -H k-1 (x) -H k (y) + H k-1 (y) = 1 {S k-1 =x} -1 {S k-1 =y} εk .
Note that for all k ≥ 1, and all x, y ∈ Z, |d k (x, y)| ≤ 1, and that

n k=1 E(d 2 k (x, y) | F k-1 ) = Var(ε) n k=1 (1 {S k-1 =x} -1 {S k-1 =y} ) 2 = Var(ε)(N n (y) + N n (x)) .
Therefore,

E (H n (x) -H n (y)) 2ν 1 2ν ≤ C 1 + E [N ν n (y)] 1 2ν + E [N ν n (x)] 1 2ν
≤ C(1 + 2n (1-1/α)/2 ) (by using (34))

≤ 3Cn (1-1/α)/2 |x -y| (α-1)/2 ,
since |x -y| ≥ 1, and n ≥ 1. Hence, according to [START_REF] Jain | Asymptotic behavior of the local time of a recurrent random walk[END_REF] (see Equation [START_REF] Chen | Moderate and small deviations for the ranges of one-dimensional random walks[END_REF]),

E ( N n (x) -N n (y)) 2ν 1 2ν ≤ pE (N n (x) -N n (y)) 2ν 1 2ν + E (H n (x) -H n (y)) 2ν 1 2ν ≤ C ν n (1-1/α)/2 |x -y| (α-1)/2 ,
for some constant C ν > 0. This proves Lemma 25.

This concludes also the proof of Lemma 24.

5.3. Expression of the return probability by an integral. According to the result of the previous subsection, we are led to the study of P( Z n = 0, S n = 0, Ω n ). As in Lemma 5, we have :

P(M n = (0, 0), Ω n ) = P( Z n = 0, S n = 0, Ω n ) = 1 2π π -π E   y∈Z ϕ ξ (t N n (y))1 {Sn=0} 1 Ωn   dt .
By following the proof of Lemma 5 (note that a priori y N n (y) is not equal to n here), we get

P( Z n = 0, S n = 0, Ω n ) = d 2π π/d -π/d E   y∈Z ϕ ξ (t N n (y))1 { y Nn(y)∈d 0 Z} 1 {Sn=0} 1 Ωn   dt. (36)
In the sequel we consider η, γ and ε satisfying all the hypotheses of Section 2.4 and γ < (α -1)/(4α).

5.4. Estimate of the integral away from the origin. The following is very similar to the case of RWRS.

Lemma 26. We have

π/d n -δ+η E   y∈Z |ϕ ξ (t N n (y))|1 Ωn   dt = o(n -1-1 αβ ).
Proof. First set

V n := y∈Z N n (y) β . Since on Ω n , y N n (y) = n k=1 ε k ≥ np/2 and N n (y) ≤ N n (y) ≤ n 1-1 α +γ
, by following the proof of Lemma 6, we get on Ω n :

V n ≥ cn δβ-γ , for some constant c > 0. Let now ε be as in Proposition 9. Then the proofs of Proposition 8 and 9 lead to

n -1+ 1 α +ε n -δ+η E   y∈Z |ϕ ξ (t N n (y))|1 Ωn   dt = o(n -1-1 αβ ).
But we can also easily adapt the proof of Proposition 10 to obtain :

π/d n -1+ 1 α +ε E   y∈Z |ϕ ξ (t N n (y))|1 Ωn   dt = o(n -1-1 αβ ).
Indeed we just need to use "flat peaks" instead of peaks. These "flat peaks" are defined as follows. Let M and N be such that P(Y where H is any multiple of M + N + 1, and one can define analogously a "lower flat peak". We leave to the reader to check that we can then follow the proof of Proposition 10 simply by replacing everywhere peaks by flat peaks. This concludes the proof of Lemma 26.

5.5. Estimate of the integral near the origin. We turn now to the estimate of the integral in (36) on the interval [-n -δ+η , n -δ+η ]. For this we will roughly follow the same lines as for the proof of Proposition 7. However the technical details are more involved here, since we have to make all calculus conditionally to {S n = 0}. The first step is the following lemma:

Lemma 27. We have

sup n E n δβ V n 1 β-1 1 Ωn S n = 0 < +∞. ( 37
) Proof. Remind that on Ω n , np/2 ≤ y N n (y) ≤ V 1/β n R 1-1/β n
. Observe on the other hand that δβ/(β -1) = β/(β -1) -1/α. Thus there is a constant C > 0 such that for all n ≥ 1, on Ω n ,

n δβ V n 1 β-1 ≤ C R n n 1 α
.

It follows from the above inequality that

E n δβ V n 1 β-1 1 Ωn 1 {Sn=0} ≤ CE R n n 1 α 1 {Sn=0} .
Set m := ⌊n/2⌋ and m ′ := ⌈n/2⌉. By using that R

n ≤ R m ′ + # {S m ′ +1 , • • • , S n } = R m ′ + # {S m ′ +1 -S n , • • • , S n-1 -S n , 0}
and Markov property (respectively on the sequences (S k , k ≥ 0) and (S n -S n-k , 0 ≤ k ≤ n)), we get [START_REF] Gall | The range of stable random walks[END_REF] Equation (7.a) p.703). We next divide all terms by P(S n = 0) which is of order n -1/α and this proves the lemma.

E n δβ V n 1 β-1 1 Ωn 1 {Sn=0} ≤ CE R m ′ n 1 α × sup x P x (S m = 0) = O n -1 α , since sup x P x (S m = 0) = O(n -1/α ) and E(R m ′ ) = O(n 1/α ) (see
We deduce the Lemma 28. We have

P( Z n = 0, S n = 0, Ω n ) = n -1-1 αβ d E n δ V 1 β n 1 { y Nn(y)∈d 0 Z} 1 Ωn S n = 0 f α (0)f β (0) + o(n -1-1 αβ ).
Proof. By following the proof of Lemma 12, we see that, uniformly on Ω n , we have:

|t|≤n -δ+η y ϕ ξ (t N n (y)) -e -|t| β Vn(A 1 +iA 2 sgn(t)) dt = o( V -1 β n ) .
By using Lemma 27, we get

|t|≤n -δ+η E y ϕ ξ (t N n (y)) -e -|t| β Vn(A 1 +iA 2 sgn(t)) 1 Ωn 1 {Sn=0} dt = o(1) × E V -1 β n 1 Ωn 1 {Sn=0} = o(n -δ-1 α ) × E (n δβ V -1 n ) 1 β-1 1 Ωn S n = 0 β-1 β (since P(S n = 0) = O(n -1 α )) , = o(n -1-1 αβ ).
By using (36) and Lemma 26, we see that it remains to estimate

|t|≤n -δ+η E e -|t| β Vn(A 1 +iA 2 sgn(t)) 1 { y Nn(y)∈d 0 Z} 1 {Sn=0} 1 Ωn dt.
But, as in the proof of Lemma 13, we have

|t|≤n -δ+η e -|t| β Vn(A 1 +iA 2 sgn(t)) dt = n -δ 2π n δ V 1 β n f β (0) + o(n -δ ),
uniformly on Ω n . We next take the expectation in both sides and we conclude the proof by using that P(S n = 0) ∼ f α (0)n -1/α .

The following lemma allows us to get rid of 1 { y Nn(y)∈d 0 Z} .

Lemma 29. Assume that d 1 is a multiple of d 0 . On {S n = 0}, we have

y N n (y) ∈ d 0 Z ⇔ n ∈ d 0 Z.
Proof. Let m n := y N n (y) = n k=1 ε k be the number of horizontal moves before time n. If S n = 0, the number n -m n of vertical moves is necessarily in d 1 Z and so in d 0 Z, since d 1 is a multiple of d 0 by hypothesis. Hence m n is in d 0 Z if and only if n is in d 0 Z.

We will need the following estimate:

Lemma 30. Let V n := x∈Z N n (x) β . Then E | V n -p β V n | S n = 0 = O(n δβ-α-1 2α ).
Proof. Set again m = ⌊n/2⌋ and m ′ = ⌈n/2⌉. By using the inequality |a β -b β | ≤ β|a -b|(a β-1 + b β-1 ) and the Cauchy-Schwarz inequality, we get

E | V n -p β V n | S n = 0 ≤ β E x∈Z ( N n (x) β-1 + p β-1 N n (x) β-1 ) 2 S n = 0 1/2 × E x∈Z ( N n (x) -pN n (x)) 2 S n = 0 1/2 . ( 38
)
We now estimate both expectations in the right hand-side of the above inequality. Then Markov property again and (34) show that

E x∈Z ( N n (x) β-1 + p β-1 N n (x) β-1 ) 2 S n = 0 ≤ CE[R m ′ ] × E N m ′ (0) 2(β-1) = O n 2(β-1)(1-1 α )+ 1 α . ( 39
)
The same argument gives

The lemma now follows by combining (38), ( 39) and ( 40) since (β -1) 1 -

1 α + 1 2α + 1 2 = δβ -α-1
2α .

Lemma 31. Conditionally to the event {S n = 0}, the sequence (V n /n δβ , n ≥ 0) converges in distribution to the random variable R (L 0 1 (x)) β dx.

Proof. According to [START_REF] Dombry | A functional approach for random walks in random sceneries Electronic[END_REF], the lemma will essentially follow from the two following statements :

(RW1) The sequence of conditioned processes n -1/α S ⌊nt⌋ | S n = 0 , t ∈ [0, 1] converges in distribution to the bridge ( U 0 t , t ∈ [0, 1]), as n → ∞. (RW2) (i) Part (RW1) is proved in [START_REF] Liggett | An invariance principle for conditioned sums of independent random variables[END_REF].

We prove now (RW2) starting with Part (i). By using the same argument as in the proof of Lemma 30, we get

E N n (y) 2 | S n = 0 ≤ C(E[N m (y) 2 ] + E[N m ′ +1 (-y) 2 ]),
for some constant C > 0, with m and m ′ as in the previous lemma. The desired result now follows from Lemma 1 in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF].

For Part (ii), set N n (x, y) := N n (x) -N n (y). Then use again the argument of the previous lemma, which gives

E[N n (x, y) 2 | S n = 0] ≤ C(E[N m (x, y) 2 ] + E[N m ′ +1 (-x, -y) 2 ] + 1),
for some constant C > 0. The result then follows from Lemma 3 in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF].

We can now apply Theorem 4.1 in [START_REF] Dombry | A functional approach for random walks in random sceneries Electronic[END_REF] in the case when the random scenery is a sequence of i.i.d. random variables with β-stable distribution and with characteristic function of the form θ → exp(-c|θ| β ). We deduce that conditionally to {S n = 0},

n -δ n k=1 ξ S k L -→ n→∞ R L 0 1 (x) dY x ,
where λ denotes the Lebesgue measure, and (U (s), s ∈ [0, 1]) is an α-stable process. In this case, it is also proved in [START_REF] Gall | The range of stable random walks[END_REF] that the constant c appearing in ( 41) is E [λ(U ([0, 1]))], so that

R n E[R n ] → λ(U ([0, 1])) E [λ(U ([0, 1]))] in distribution. ( 43
)
Our aim in this appendix is to prove the following result:

Lemma 33. Assume that α ∈ (0, 2]. Let γ ∈ (0, 1/α), and set

R n := E[R n ]n -γ ≤ R n ≤ E[R n ]n γ .
Then there exists a constant C > 0, such that

P(R n ) = 1 -O(exp(-Cn γ )) . ( 44
)
Proof. We first prove that for n large enough,

P [R n ≥ E[R n ]n γ ] ≤ exp(-Cn γ ). ( 45
)
Let us recall that for every a, b ∈ N, we have

P(R n ≥ a + b) ≤ P(R n ≥ a)P(R n ≥ b) . ( 46
)
The proof is given for instance in [START_REF] Chen | Moderate and small deviations for the ranges of one-dimensional random walks[END_REF] and goes as follows. Let τ := inf {k : R k ≥ a}. Note that τ is a stopping time, and that R τ = a on {τ < ∞}. Moreover, proving (46). Hence,

P
P (R n ≥ E[R n ]n γ ) ≤ P R n ≥ ⌊3E[R n ]⌋ n γ 3 ≤ P (R n ≥ ⌊3E[R n ]⌋) n γ 3 ≤ E[R n ] ⌊3E[R n ]⌋ n γ 3 ≤ E[R n ] 3E[R n ] -1 n γ 3 ≤ 1 2 n γ 3 .
This finishes the proof of (45). It remains now to prove that for n large enough,

P R n ≤ E[R n ]n -γ ≤ exp(-Cn γ ). ( 47
)
To this end, let I 1 , • • • , I N be disjoint subsequent intervals of {0, • • • , n}, of the same length l n depending on n, so that l n ≫ 1 and N = ⌊n/l n ⌋. Note that

R n ≥ N max j=1 (# {S k , k ∈ I j }) ,
and that the random variables (# {S k , k ∈ I j } , 1 ≤ j ≤ N ) are i.i.d with the same law as R ln . Hence

P R n ≤ E[R n ]n -γ ≤ P N max j=1 (# {S k , k ∈ I j }) ≤ E(R n )n -γ = P R ln ≤ E[R n ]n -γ N . (48)

1. 2 .

 2 The results. Our first statement is obtained in the case when the ξ ′ s are Z-valued random variables. Let ϕ ξ (u) := E[e iuξ 1 ] be the characteristic function of ξ 1 . Remember that there exists an integer d ≥ 1 such that {u : |ϕ ξ (u)| = 1} = 2π d Z (d is the g.c.d. of the set of b -c where b and c belong to the support of the distribution of ξ 1 ) 1 .

2. 3 .

 3 The event Ω n . Set N * n := sup y N n (y) and R n := #{y : N n (y) > 0} .

C- 1 : 1 α

 11 n (y) := # k = 0, . . . , n T S kT = y and (X kT +1 , . . . , X (k+1)T ) = C ± , be the number of peaks based on y before time n (and at times which are multiple of T ), and letp n := #{y ∈ Z : C n (y) ≥ n 1-1 α -2γ },be the number of sites y ∈ Z on which at least n 1-1 α -2γ peaks are based.Lemma 15. On Ω n ∩ D n , we have p n ≥ 3N M n -αγ , for n large enough.

5. 2 .

 2 The event Ω n . Let (N n (y), y ∈ Z) and R n denote respectively the local time process and the range of S at time n:N n (y) := #{k = 0, ..., n -1 : S k = y} and R n := # {y : N n (y) > 0} .

1 =

 1 N ) > 0 and P(Y 1 = -M ) > 0. Then an "upper flat peak" is a sequence of the type (Y H+1 , . . . , Y H+M , ε H+M +1 , Y H+M +2 , . . . , Y H+M +N +1 ) = (N, . . . , N, 1, -M, . . . , -M ),

  First note that N n (x) = N m (x)+(N n (x)-N m (x)) and that the sequence ((N n (x)-N m (x), x ∈ Z) | S n = 0) has the same distribution as ((N m ′ +1 (-x) -N 1 (-x), x ∈ Z) | S n = 0). Thus the Markov property gives E x∈Z ( N n (x) β-1 + p β-1 N n (x) β-1 ) 2 S n = 0 ≤ 4 x∈Z E N n (x) 2(β-1) S n = 0 ≤ C x∈Z M ∈Z E N m (x) 2(β-1) 1 {Sm=M } P(S m ′ = -M ) P(S n = 0) + x∈Z M ∈Z E (N m ′ (x)) 2(β-1) 1 {S m ′ =-M } P(S m = M ) P(S n = 0) ,for some constant C > 0. Since sup M P(S m ′ = -M )/P(S n = 0) < +∞ and sup M P(S m = M )/P(S n = 0) < +∞, we getE x∈Z ( N n (x) β-1 + p β-1 N n (x) β-1 ) 2 S n = 0 ≤ C x∈Z E N m ′ (x) 2(β-1) .

x∈ZE(

  N n (x) -pN n (x)) 2 S n = 0 ≤ C x∈Z E ( N m (x) -pN m (x)) 2 + x∈Z E ( N m ′ (x) -pN m ′ (x)) 2 ,for some constant C > 0. Then by using (35) (note that εk is centered and independent of(S ℓ-1 , ε ℓ , S k-1 ) if ℓ < k), we get x∈Z E ( N n (x) -pN n (x)) 2 S n = 0 = O(n).

yEE N n ⌊n 1 α x⌋ -N n ⌊n 1 α y⌋ 2 S n = 0 ≤

 1120 N n (y) 2 | S n = 0 = O(n 2-2 α ).(ii) There exists a constant C > 0 such that for every x, y ∈ R, Cn 2-2 α |x -y| α-1 .

  (R n ≥ a + b) = P(τ ≤ n; R n ≥ a + b) = n j=1 P(τ = j; R n ≥ R j + b) Now, for j ≤ n, R n ≤ R j + # {S j+1 , • • • , S n } = R j + # {S j+1 -S j , • • • , S n -S j }.By independence, we have P(R n ≥ a + b) ≤ n j=1 P(τ = j)P(R n-j ≥ b) ≤ P(R n ≥ b)P(τ ≤ n),

ANR projects MEMEMO and RANDYMECA.

where (Y x , x ∈ R) is a two-sided β-stable Lévy process independent of U 0 and limit in distribution of n -1 β ⌊nx⌋ k=0 ξ k , x ∈ R , when n → ∞. Therefore (see for instance Lemma 5 in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF]), for every θ ∈ R,

which proves the lemma.

Lemma 32. Conditionally to the event {S n = 0}, the sequence

Proof. By Lemma 31, the sequence (n δβ V -1 n , n ≥ 0) converges in distribution to |L 0 | -β β , conditionally to {S n = 0}. On the other hand, Lemma 24 implies that the sequence (1 Ωn , n ≥ 0) converges in distribution to the constant 1, conditionally to {S n = 0}. Hence, the sequence

Therefore, since γ < (α -1)/(4α), the left hand side in the above equation converges to 0 when n → ∞. The lemma follows.

We finally obtain the Proof of Theorem 20. The uniform integrability of the sequence (n δ V -1/β n 1 Ωn , n ≥ 0) conditionally to {S n = 0} is deduced from Lemma 27 . It then follows from Lemma 32 that

The theorem now follows from Lemmas 28 and 29.

Appendix A. Control of the range

We first gather some known facts about the range R n of the random walk (S n , n ≥ 0). First of all, this walk is transient if, and only if, α < 1. Moreover, there exists a constant c > 0 such that [START_REF] Gall | The range of stable random walks[END_REF] Theorem 6.9 p.698) , n 1/α if α > 1( see [START_REF] Gall | The range of stable random walks[END_REF] Equation (7.a) p.703) .

(41)

In addition, if α ≤ 1 (see [START_REF] Spitzer | Principles of random walks[END_REF] p.38-40 for α < 1, and [27] Theorem 6.9 for α = 1), then

. By (41), this gives

(49)

For n large enough, E[R ln ] ≥ 2E[R n ]n -γ , and it follows from (48) that

tends to zero by (42). By (43), for α > 1, we have lim sup

))] < 1, since a.s. λ(U ([0, 1])) > 0. In any case there exists p < 1, such that for all γ ∈ (0, 1/α), and for n large enough,

Together with (50) and (49), this proves (47) and the lemma.


Pierre Lubin
email: lubin@enscbp.fr

Stéphane Glockner
email: glockner@enscbp.fr

Hubert Chanson
email: h.chanson@uq.edu.au

Numerical simulation of a weak breaking tidal bore

Keywords: Environmental flows, Navier-Stokes equations, numerical simulations, tidal bores

A tidal bore is a natural and fragile phenomenon, which is of great importance for the ecology of an estuary. The bore development is closely linked with the tidal range and the river mouth shape, and its existence is sensitive to any small change in boundary conditions. Despite their ecological and cultural value, little is known on the flow field, turbulent mixing and sediment motion beneath tidal bores. Indeed, some striking features can be highlighted in two-dimensional simulations, such as large velocity fluctuations and flow recirculation structures. Using Large Eddy Simulation method, we present numerical results that show the complicated turbulent structures and their unsteadiness under a tidal bore.

Introduction

A tidal bore is a positive surge propagating upstream as the tidal flow turns to rising in river mouths exhibiting converging funnelled channel forms during low freshwater conditions. The tidal bore is a very vulnerable process and it results from a fragile balance between many parameters (e.g. bathymetry, tidal conditions, etc.). Very few field observations were conducted to date, while some recent laboratory experiments brought new insights to the turbulent motion (Koch andChanson, 2005, 2009). Nevertheless, the tidal processes remain poorly understood today [START_REF] Chanson | Current knowledge in hydraulic jumps and related phenomena. a survey of experimental results[END_REF]. A recent numerical model based upon the Navier-Stokes equations [START_REF] Furuyama | A numerical study of open channel flow hydrodynamics and turbulence of the tidal bore and dam-break flows[END_REF]) was compared to laboratory experiments [START_REF] Koch | An experimental study of tidal bores and positive surges: Hydrodynamics and turbulence of the bore front[END_REF]. Some interesting features were observed, but the results lacked a fine mesh grid resolution and accurate numerical schemes.

The goal of our work is to simulate this unsteady two-phase flow tidal bore motion using Large Eddy Simulation method to gain a further understanding of the tidal bore processes. We aim at describing accurately the free-surface behavior and the turbulent flow structure.

Numerical model

On a fixed orthogonal curvilinear grid, an incompressible multiphase phase flow between non-miscible fluids can be described by the Navier-Stokes equations in their multiphase form. The governing equations for the Large Eddy Simulation (LES) of an incompressible fluid flow are classically derived by applying a convolution filter to the unsteady Navier-Stokes equations.

The velocity / pressure coupling is solved with a pressure correction method [START_REF] Goda | A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows[END_REF]. The space derivatives of the inertial term are discretized by a hybrid Upwind-Centered scheme and the viscous term is approximated by a second order centered scheme [START_REF] Lubin | Threedimensional large eddy simulation of air entrainment under plunging breaking waves[END_REF]. The interface tracking is achieved by a Volume Of Fluid method (VOF): a Lax-Wendroff TVD scheme (Total Variation Diminishing) is used to solve directly the free-surface evolutions. A dual grid, or underlying grid (Rudman, 1998), is used to gain an improved accuracy for the interface description, the mesh grid size being divided by two in each direction for the interface tracking. The turbulent viscosity is calculated with the Mixed Scale model [START_REF] Sagaut | Large Eddy Simulation for incompressible flows -An introduction[END_REF]. The numerical model has proved its accuracy for coastal applications [START_REF] Lubin | Threedimensional large eddy simulation of air entrainment under plunging breaking waves[END_REF] and has been benchmarked through numerous test-cases including mesh refinement analysis [START_REF] Lubin | Large eddy simulation of plunging breaking waves[END_REF]. The time discretization is implicit and the equations are discretized on a staggered grid thanks to a finite volume method. The MPI library HYPRE is used to solve the linear system of the prediction and correction steps [START_REF] Falgout | The design and implementation of hypre, a library of parallel high performance preconditioners[END_REF].

Numerical configuration of a breaking tidal bore

The experimental configuration consists in the generation of a weak positive surge by a rapid partial gate closure at the downstream end of the control volume and its upstream propagation against the initially steady flow (Koch andChanson, 2005, 2009). The numerical configuration consists in an initial rectangular steady flow motion (from the right side of the numerical domain to the left side) with an initial steady velocity (V 0 = 1.021 m.s -1 ). The initial water depth is d 0 = 0.0785 m, as presented in Figure 1. The twodimensional numerical domain is 10 m long and 1 m high and is discretized

Results

The initial rectangle of water hits the left wall, the water runs-up the wall and splashes down. The generated bore then propagates upstream, towards the right side of the numerical domain. The initial large free-surface deformations are in accordance with the experimental photographs [START_REF] Koch | An experimental study of tidal bores and positive surges: Hydrodynamics and turbulence of the bore front[END_REF]. The celerity of the bore front is approximatively 0.5 m.s as experimentally recorded. We investigated the horizontal and vertical velocity components as functions of time (not shown here). The bore front passage is associated with a rapid flow deceleration, coupled with a sudden increase in water depth. Some flow reversal is observed next to the bed.

This was discussed and associated with some transient flow separation and recirculation (Koch andChanson, 2005, 2009). Some large structures are shown in Figure 2: a main recirculation structure observed beneath the bore front and propagating upstream (towards the right side of the numerical domain), and some macroturbulent structures separating for the previous one and propagating downstream (towards the left side of the numerical domain). The height of these large eddies is approximately half of the downstream water depth, at the time of their generation.

It can be observed that the structures are generated in sequence below the bore front, the continuous process exhibiting some regular temporal and spatial frequencies. The main large flat-shaped recirculation cell appears as the bore front propagates. It then grows in size as the water depth suddenly increases. This cell then splits in structures of similar size. The recirculation structures then separate regularly in time and are released to propagate downstream, spaced every 20 cm from each other. The eddies are advected at about V adv ≃ 0.3 m.s -1 , which gives a ratio V adv /V 2 ≃ 0.6, V 2 being the flow velocity after the bore passage. These vortical structures remain next to the bed as these persisting coherent structures are advected towards the left side of the numerical domain while the breaking bore propagates upstream. This implies that a great amount of sediment could be placed into suspension and transported by the main flow. These original results confirm and illustrate the experimental observations identifying for the first time these recirculations under tidal bores (Koch andChanson, 2005, 2009).

At this point, these preliminary results are limited by two aspects. First, the numerical configuration is two-dimensional. Then, there is a need for realistic unsteady inflow conditions to be specified at the inlet boundary. Some numerical tests were performed with the basic technique consisting in generating turbulent inflow data by taking the mean experimental velocity profile with superimposed random fluctuations. The generated data do not exhibit any spatial or temporal correlations, and the pseudo turbulence is quickly dissipated. An effective method to generate synthetic eddies on the inlet plane is under implementation [START_REF] Jarrin | A syntheticeddy-method for generating inflow conditions for large-eddy simulations[END_REF].

Conclusion and future work

The major result of this ongoing work is the identification of recirculation structures generated in sequence below the front of the propagating tidal bore and advected downstream. The main features of the flow are in accordance with the basic experimental results (weak breaking bore, flow reversal and rapid flow deceleration). Some 3D numerical developments are undertaken to overcome the limitations of the inlet boundary conditions, to confirm these first observations and to investigate more into details the generation process of the recirculation structures.

Acknowledgement

The authors wish to thank the Aquitaine Regional Council for the financial support dedicated to a 256-processor cluster investment, located in the TREFLE laboratory. This work was performed using HPC resources from GENCI-CINES (Grant 2009-c2009026104). Prof. Hubert Chanson also acknowledges the funding for a visiting professorship position at the Université de Bordeaux and a fellowship from the Australian Academy of Science.

Figure 1 :

 1 Figure 1: Sketch of the initial conditions for the tidal bore generation, immediately after the gate closure corresponding to the wall boundary condition set at the left side of the numerical domain.

  -1 , a. t = 12.72 s b. t = 14.45 s

Figure 2 :

 2 Figure 2: Streamlines indicating recirculation structures under the propagating tidal bore.


F Zomer
email: zomer@lal.in2p3.fr

Laser beam-electron Compton scattering: technology and applications

The goal of this presentation is to introduce the properties and scientific interests of the laser beam-electron beam Compton scattering. There are many possible applications for this process but the main technical difficulty is the requested very high laser beam average power. The various solutions that are presently worked out to provide the requested average power will be reviewed as well as the various electron accelerator technologies. Finally, the various Compton machine operating at present and the future projects will be described.

General Introduction

Laser electron scattering (see Fig. 1) is a well known process which can provide monochromatic X or gamma ray beams [1]. Such beams can be used in a large application domain. In this presentation we will cover the following topics:

Properties of the Compton scattering. Applications. Optical devices needed for Compton X/gamma ray source.

Accelerator devices needed for Compton X/gamma ray source. Status of the Compton project/installation X/gamma ray sources.

Properties of Compton scattering

The diffusion of an electromagnetic plane wave by an electron at rest, with mass m e and charge q, is a process known as 'Thomson scattering' [START_REF] Jackson | Classical electrodynamics[END_REF]. From a classical point of view, the process can be visualised as a plane wave of frequency ω 0 impinging on a charge. The scattered wave is not plane but, in the approximation where the recoil effect can be neglected (i.e. m e >>ω 0 ) the electromagnetic wave frequency is conserved. However, the very general formula are derived within quantum electrodynamics [START_REF] Berestetskii | Quantum electrodynamics[END_REF] where the scattering process is understood as the succession of the absorption and the emission of the photon by the electron and is known as 'Compton scattering'. We shall adopt this latter corpuscular point of view which includes electron recoil effect and polarisation properties [START_REF] Berestetskii | Quantum electrodynamics[END_REF].

The interest of Compton scattering is related to the kinematical properties of the X/gamma rays. First, the energy distribution exhibits a kinematical limit given by E max »4γ 2 ω 0 with γ =E e /m e where E e is the electron incoming energy. Since m e »0.5MeV one sees that, comparing to synchrotron radiation process [1], the Compton scattering is a very efficient way to boost the laser photon energy. Second, Compton scattering is a two body process so that the angle and energy on the produced X/gamma rays are correlated and θ=0 (see Fig. 1) corresponds to the maximum scattered energy. With a simple collimation system, one can thus produce a quasi monochromatic X/gamma ray beam. Third, depending on the laser beam polarisation, and even for an unpolarised electron beam, one can control the polarisation of the scattered X/gamma rays.

In summary, Compton scattering offers the possibility to produce a quasi monochromatic polarised X/gamma ray beam.

In the presentation, we shall review these kinematical properties and provide illustrative examples.

Applications of Compton scattering

In the presentation, we shall describe the vast domain of applications of Compton scattering. We shall distinguish three kinematical depending on the electron beam energy E e :

For E e ~ 10-50 MeV, X rays of few keV to 100keV are produced. The main application domains are: medical (radiography and radiotherapy), material science, art and palaeontology studies, and pharmacology. In this field, the goal is to provide compact electron accelerators that can be located inside an hospital or a museum.

For E e ~ 300 MeV, gamma rays of few MeV energy are produced which can be used in nuclear science and nuclear survey.

For E e > 1 GeV, high energy photons are produced that can be for electron beam diagnostics (laser wire, polarimeter), to produce polarised positrons and to create high energy photon beams. We shall describe some of these applications during the presentation.

Optical devices needed for Compton X,gamma rays source

The main drawback of Compton scattering process is its small cross section. It means that one must use special optical amplification schemes in order to reach the X/gamma rates required by the applications described in the previous section. In the presentation, we shall describe two amplification schemes that use optical resonators (see Fig. 2): the external Fabry-Perot cavity and a recirculation cavity. The expected performance of the external Fabry-Perot cavity system will be described in details.

Accelerator devices needed for Compton X/gamma ray source

One distinguishes two classes of electron accelerators: the linear accelerators (LINAC) and ring accelerators. Among the LINACs, one further distinguishes those which use supraconducting accelerating cavities for those using warm cavities.

In the presentation, we shall give the advantages and disadvantages of each electron accelerator configurations. We shall also briefly describe the different electron beam dynamics corresponding to these machines.

Status of project/installation

There are presently Compton machines operating in China, Japan and USA. We shall describe these installations during the presentation, as well as the new projects which recently appeared in Europe, Japan and USA.

References Informations

Fig. 1 .

 1 Fig. 1. Schematic view of the laser-electron Compton Scattering.

Fig. 2 .

 2 Fig. 2. Two optical scheme used for increasing the laser average power: (a) external Fabry-Perot, (b) Ring cavity recirculation.