Brice Franke

Françoise Pène
email: francoise.pene@univ-brest.fr

Martin Wendler
email: martin.wendler@rub.de

CONVERGENCE OF U-STATISTICS INDEXED BY A RANDOM WALK TO STOCHASTIC INTEGRALS OF A LÉVY SHEET

Keywords: random walk, random scenery, U -statistics, stable limits, Lévy sheet Françoise

A U -statistic indexed by a Z d 0 -random walk (Sn)n is a process Un := n i,j=1 h(ξ S i , ξ S j ) where h is some real-valued function and (ξ k ) k is a sequence of iid random variables, which are independent of the walk. Concerning the walk, we assume either that it is transient or that its increments are in the normal domain of attraction of a strictly stable distribution of exponent α ∈ [d 0 , 2). We further assume that the distribution of h(ξ 1 , ξ 2 ) belongs to the normal domain of attraction of a strictly stable distribution of exponent β ∈ (0, 2). For a suitable renormalization (an)n we establish the convergence in distribution of the sequence of processes (U ⌊nt⌋ /an)t; n ∈ N to some suitable observable of a Lévy sheet (Zs,t)s,t. The limit process is the diagonal process (Zt,t)t when the walk is transient or when α = d 0 . When α > d 0 = 1 the limit process is some stochastic integral with respect to Z.

Introduction

Given a random walk (S n ) n≥0 on Z d0 and a sequence of independent identically distributed (iid) real random variables (ξ k ) k∈Z d 0 , independent one from each other, one can consider the random walk in random scenery S n := n k=1 ξ S k . In particular one is interested in the limit behavior of the sequence of renormalized processes (ν -1 n S [nt] ) t≥0 ; n ∈ N. In this context the following assumptions are usually made: (A) either S n is transient or there exists some α ∈ [d 0 , 2] such that n -1 α S n ; n ∈ N converges in distribution to a random variable; (B) n -1 β n k=1 ξ k ; n ∈ N converges in distribution to a random variable for some β ∈ (0, 2]. Note that in the case α > d 0 = 1 the assumption (A) implies that the sequence of stochastic processes (n -1 α S ⌊nt⌋ ) t>0 ; n ∈ N converges in distribution to some α-stable Lévy process (Y t ) t>0 which admits a local time (L t (x), t ≥ 0, x ∈ R). Similarly, assumption (B) implies that (n -1 β ⌊nt⌋ k=1 ξ k ) t>0 ; n ∈ N converges in distribution to some β-stable process (Z t ) t>0 . 1 Subsequently we will use (Z -t ) t>0 to denote an independent copy of (Z t ) t>0 .

Random walks in random scenery have been studied by many authors since the early works of Borodin [START_REF] Borodin | A limit theorem for sums of independent random variables defined on a recurrent random walk[END_REF][START_REF] Borodin | Limit theorems for sums of independent random variables defined on a transient random walk[END_REF] and Kesten and Spitzer [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF]. In particular, [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF][START_REF] Deligiannidis | An asymptotic variance of the self-intersections of random walks[END_REF][START_REF] Castell | Limit theorems for one and two-dimensional random walks in random scenery[END_REF]] complete the study of the limit in distribution of random walks in random scenery. The asymptotic behavior of the sequence (ν -1 n S ⌊nt⌋ ) t>0 ; n ∈ N is summarized in the following table (where d 1 and d 2 are explicit constants depending on (S n ) and on β):

Cases normalization Limit process Space of convergence in distribution transient ν n := n 1 β (d 1 Z t ) t finite distributions if β = 1:Skorokhod space with M 1 -metric α = d 0 ν n := n 1 β (log n) 1-1 β (d 2 Z t ) t finite distributions if β = 1:Skorokhod space with M 1 -metric α > d 0 ν n := n 1-1 α + 1 αβ (∆ t := R * L t (x) dZ x ) t
Skorokhod space with J 1 -metric

In this paper we want to do a similar investigation for U-statistics indexed by a random walk. To introduce the objects let E be some measurable set and (ξ k ) k∈Z d 0 an iid sequence of E-valued random variables. Often we might abbreviate this family of random variables by ξ and call it the scenery. Moreover, let (S n ) n≥1 be as above a random walk on Z d0 , which is independent of the scenery ξ. We will also use the short notation S for the random walk. For some measurable function h : E 2 → R, we consider the U -statistic indexed by S defined through

U n := n i,j=1
h(ξ Si , ξ Sj ).

We are interested in results of distributional convergence for (U n ) n (after some suitable normalization) under the assumption that the distribution of h(ξ 1 , ξ 2 ) is in the normal domain of attraction of a β-stable distribution. Let us assume without loss of generality that h is symmetric.

If β > 1 we can introduce ϑ k := E[h(ξ 0 , ξ k )|ξ 0 ]. Two different situations can occur. We will say that the kernel is degenerate if ϑ 1 = 0 almost surely. Otherwise, we will say that the kernel is non-degenerate.

The case when when h(ξ 1 , ξ 2 ) is square integrable and centered (which implies β = 2) has been fully studied by Guillotin-Plantard and her co-authors. In this case only two kind of behaviors can occur:

(a) the kernel is non-degenerate, then one can use Hoeffding decomposition to show that U n behaves essentially as n i,j=1 (ϑ Si + ϑ Sj ) = 2n n i=1 ϑ Si . (b) the kernel is degenerate, then Hilbert-Schmidt theory can be used to represent the kernel as h(x, y) = p λ p φ p (x)φ p (y) and to show that U n behaves as p λ p ( n i=1 φ p (S i )) 2 . This has been proved by Cabus and Guillotin-Plantard in [START_REF] Cabus | Functional limit theorems for U-statistics indexed by a random walk[END_REF] for random walks in Z d0 with d 0 ≥ 2 and by Guillotin-Plantard and Ladret in [START_REF] Guillotin-Plantard | Limit theorems for U-statistics indexed by a one dimensional random walk[END_REF] for random walks in Z.

Note that the situation treated in [START_REF] Cabus | Functional limit theorems for U-statistics indexed by a random walk[END_REF] splits into the case d 0 > 2, where the walk is transient, and the singular case d 0 = 2, where the random walk is null recurrent. However, in this last case the limit process (Y t ) t≥0 does not have local time. In contrast to this the assumptions made in [START_REF] Guillotin-Plantard | Limit theorems for U-statistics indexed by a one dimensional random walk[END_REF] correspond to some null recurrent random walk with existing local time for (Y t ) t≥0 ; i.e.: α > d 0 = 1.

The special form of the representations given in (a) and (b) implies that for β = 2, the study of (U n ) n can be reduced to the study of some suitable random walk in random scenery (either n i=1 ϑ Si or n i=1 φ p (S i )). Thus the limits can be expressed in terms of processes which already occurred in the random scenery situation.

In the transient case or if d 0 = 2 the limit process turns out to be Brownian motion (B t ) t≥0 when the kernel is non-degenerate. In the degenerate situation the limit has the representation p λ p (B (p) t ) 2 , where (B (p) t ) t≥0 ; p ∈ N is a sequence of independent Brownian motions (see [START_REF] Cabus | Functional limit theorems for U-statistics indexed by a random walk[END_REF]). If on the other hand α > d 0 = 1, then in the non-degenerate situation the limit is the usual process ∆ t := R * L t (x) dB x , where (B x ) x>0 and (B -x ) x>0 are independent one-dimensional Brownian motions. In the degenerate case the limit takes the form p λ p R * L t (x) dB

(p) x 2
, where the pairs (B (p)

x ) x>0 , (B (p) -x ) x>0 form a sequence of independent copies of the pair (B x ) x>0 , (B -x ) x>0 (see [START_REF] Guillotin-Plantard | Limit theorems for U-statistics indexed by a one dimensional random walk[END_REF]). Let us further mention that (a) includes the case where h(x, y) = g(x) + g(y) and that (b) includes the case when h(x, y) = g(x)g(y). Here g : E → R is a measurable function such that g(ξ 1 ) is square integrable and centered. This holds for example if h(x, y) = g(x) + g(y). The limit then turns out to be β-stable Lévy process (Z t ) t≥0 when the walk is transient or when α = d 0 . However, when α > d 0 the limit has the representation ∆ t := R * L t (x) dZ x , where (Z x ) x>0 and (Z -x ) x>0 are independent one-dimensional β-stable Lévy-motions (see [START_REF] Franke | Stable limit theorem for U-Statistic processes indexed by a random walk[END_REF]).

On the other hand in the degenerate case, when ϑ 1 = 0, different limits than those described in (b) can arise when 0 < β < 2. This is the purpose of the present paper. The limit we obtain is the diagonal process (Z (t,t) ) t≥0 of a Lévy sheet (Z t,s ) t,s≥0 , when the walk is transient or when α = d 0 , and a stochastic integral R 2 L t (x)L t (y)dZ x,y with respect to four independent copies of the Lévy sheet introduced above, when α > d 0 . These limits can be understood as two-dimensional analogues of the known limits for random walk in random scenery found by Kesten and Spitzer (see [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF]).

To be more precise, let us keep assumption (A) but replace (B) on (ξ k ) k by the following assumption on (h(ξ k , ξ ℓ )) k,ℓ :

(B') (n -1 β n k=1 h(ξ 2k , ξ 2k+1
)) n converges in distribution to a random variable with β ∈ (0, 2). This implies that if (h i,j ) i,j is a sequence of iid random variables with the same distribution as h(ξ 1 , ξ 2 ), then the sequence of stochastic processes (n -2 β ⌊nt⌋ k=1 ⌊ns⌋ ℓ=1 h i,j ) t>0 ; n ∈ N converges in law to some βstable Lévy sheet (Z s,t ) s,t>0 (which we extend on R 2 ).

In the present paper, under assumption (B') and some additional assumptions, we prove limit theorems for the U -statistic which are summarized in the following table:

Cases normalization Limit process Space of convergence in distribution transient ν 2 n = n 2 β (d 2 1 Z t,t ) t finite distribution α = d 0 ν 2 n = n 2 β (log n) 2-2 β (d 2 2 Z t,t ) t finite distribution α > d 0 ν 2 n = n 2-2 α + 2 αβ ( R 2 L t (x)L t (y) dZ x,y ) t Skorokhod space with J 1 -metric
The present paper is organized as follows. The assumptions and main results are stated in Section 2. We give some examples which satisfy our assumptions in Section 3. We prove our results concerning convergence of finite distribution in Section 4. In the spirit of [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF], our proof relies on the convergence of a suitably defined point process to a Poisson point process which is established by the use of Kallenberg theorem. In Section 5, we prove the tightness for the J 1 -metric when α > d 0 . We complete our article with some facts on the β-stable Lévy sheet Z in Appendix A. In particular a construction of stochastic integrals with respect to Z is given.

Main results

Let (Ω, F , P) be a suitable probability space and let S = (S n ) n≥0 be a Z d0 -valued random walk on (Ω, F , P) with S 0 = 0 such that one of the following conditions holds:

• the random walk (S n ) n≥0 is transient, • the random walk (S n ) n≥0 is recurrent and there exists α ∈ [d 0 , 2] such that (n -1 α S n ) n≥1
converges in distribution to a random variable Y . In this case we further assume that ∀x ∈ Z d0 , ∃n ∈ N :

P(S n = x) > 0.
Recall that, in the second case, (n -1 α S ⌊nt⌋ ) t>0 ; n ∈ N converges in distribution to an α-stable process (Y t ) t>0 such that Y 1 has the same law as Y .

In order to get a uniform notation for the different situations, we define α 0 to be a number, which is one when the random walk is transient, and which takes the value α d0 in the recurrent case. Let ξ = (ξ ℓ ) ℓ∈Z d 0 be a family of iid random variables on (Ω, F , P) with values in some measurable space E. We assume that the two families S and ξ are independent. Let h : E × E → R be a measurable function. We are interested in the properties of the U-statistics process U n := n i,j=1 h(ξ Si , ξ Sj ). In this work, we assume moreover that the following properties are satisfied. Assumption 1. Let β ∈ (0, 2).

(i) for every x ∈ E, h(x, x) = 0; (ii) h symmetric (i.e. h(x, y) = h(y, x) for every x, y ∈ E);

(iii) there exist c 0 , c 1 ∈ [0, +∞) with c 0 + c 1 > 0 such that (1) ∀z > 0, P(h(ξ 1 , ξ 2 ) ≥ z) = z -β L 0 (z), with lim z→+∞ L 0 (z) = c 0 ; and (2) ∀z > 0, P(h(ξ 1 , ξ 2 ) ≤ -z) = z -β L 1 (z), with lim z→+∞ L 1 (z) = c 1 ; (iv) there exist C 0 > 0 and γ > 3β 4 such that (3) ∀z, z ′ ∈ (0, +∞), P |h(ξ 1 , ξ 2 )| ≥ z and |h(ξ 1 , ξ 3 )| ≥ z ′ ≤ C 0 max(1, z) max(1, z ′ ) -γ ; (v) If β > 1, then E[h(ξ 1 , ξ 2 )] = 0; (vi) If β ≥ 4/3, there exists C ′ 0 > 0 and θ ′ > 3β 4 -1 such that ∀M, M ′ ∈ (0, +∞), |E [h M (ξ 1 , ξ 2 )h M ′ (ξ 1 , ξ 3 )]| ≤ C ′ 0 (M M ′ ) -θ ′ , where h M (x, y) := h(x, y)1 {|h(x,y)|≤M} + β β-1 (c 0 -c 1 )M 1-β . (vii) If β = 1, then c 0 = c 1 and lim M→+∞ E[h(ξ 1 , ξ 2 )1 {|h(ξ1,ξ2)|≤M} ] = 0.
Some examples satisfying the above assumptions are presented in the next section.

Remark 2. The following comments on the different points in Assumptions 1 might be of some help:

• Item (i) can be relaxed as will be proved in Proposition 7 below.

• Item (ii) is not restrictive since one can always replace h(z, z ′ ) by (h(z, z ′ ) + h(z ′ , z))/2 without changing the sequence (U n ) n . • Note that Item (iv) is a condition which ensures that the tail behavior resulting from coupling of the pairs (ξ 1 , ξ 2 ) and (ξ 1 , ξ 3 ) does not interfere with the tail behavior of the single terms h(ξ 1 , ξ 2 ).

A condition with the same spirit is condition (2.1) in [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF]. • If Item (iii) holds and if for every x ∈ E the distribution of h(x, ξ 1 ) is symmetric, then Item (vi) and Item (vii) are also satisfied. Indeed, in this case, c 0 = c 1 and

E [h M (ξ 1 , ξ 2 )h M ′ (ξ 1 , ξ 3 )] = E E[h(x, ξ 2 )1 {|h(x,ξ2)|≤M} ]E[h(x, ξ 2 )1 {|h(x,ξ2)|≤M ′ } ] dP ξ1 (x) = 0.
• Note that Item (iii) and Item (v) imply that the law of h(ξ 1 , ξ 2 ) is in the domain of attraction of a β-stable law for some β ∈ (0, 2).

Let (h i,j ) i,j be a sequence of iid random variables with same distribution as h(ξ 1 , ξ 2 ). Observe that the Items (i), (iii), (v) and (vii) in Assumption 1 describe the classical situation, where the sequence of random fields (n

-2 β ⌊nx⌋ i=1 ⌊ny⌋ j=1 h i,j ) x,y>0 ; n ∈ N converges in law to a β-stable Lévy sheet ( Zx,y ) x,y≥0 such that the characteristic function of Zx,y is given by E[e iz Zx,y ] = Φ xy(c0+c1),xy(c0-c1),β (z), with (4) Φ A,B,β (z) := exp -|z| β +∞ 0 sin t t β dt A -iB sgn(z) tan πβ 2 if β = 1 and (5) Φ A,B,1 (z) := exp -|z| π 2 A + iB sgn(z) log |z|
(see [13, p. 568-569]). In order to construct a continuation of the Lévy sheet Z to all of R 2 we use four independent copies

Z (ε,ε ′ ) (with ε, ε ′ ∈ {1, -1}) of Z to introduce Z x,y := Z (sgn(x),sgn(y)) |x|,|y|
for all (x, y) ∈ R 2 . In the following we will need to integrate some continuous compactly supported function ψ with respect to Z, i.e.: R 2 ψ(x, y) dZ x,y .

More information on Lévy sheets and on the construction of the integral can be found in Appendix A.

When α > d 0 = 1, we assume moreover that (Z x,y ) x,y is independent of the α-stable process (Y t ) t .

If the random walk is transient, we write N ∞ for the total number of visits of the two sided random walk (S n ) n∈Z to zero; i.e.:

N ∞ := n∈Z 1 {Sn=0} . Theorem 3 (Transient case). Suppose (S n ) n≥0 is transient and Assumption 1. We set a n := n 2 β .
Then the finite distributions of ((U ⌊nt⌋ /a n ) t>0 ) n converge to the finite distributions of (K

2 β β Z t,t ) t>0 , with K β := E[N β-1 ∞ ].
In particular the previous theorem holds for the deterministic Z-valued walk S n = n (for which K β = 1). In that case our result boils down to a result on classical U-statistics which was established by Dabrowski, Dehling, Mikosch and Sharipov in [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF]. We emphasize this point in the following corollary, since the link to the Lévy sheet was not mentioned in [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF].

Corollary 4 (Deterministic case). Suppose Assumption 1 and set a n := n 2 β . The finite distributions of (( ⌊nt⌋ i,j=1 h(ξ i , ξ j )/a n ) t>0 ) n converge to the finite distributions of (Z t,t ) t>0 . As usual Γ will stand for the Gamma function. We also write N n (x) for the occupation time of S at x up to time n, i.e.:

N n (x) := n i=1 1 {Si=x} .
We define the maximal occupation time of S up to time n through N * n := max x N n (x) and the range of S up to time n by R n := #{y ∈ Z d0 : N n (y) > 0}. We recall that, when α = d 0 , there exists c 3 > 0 such that [START_REF] Cabus | Functional limit theorems for U-statistics indexed by a random walk[END_REF] R n ∼ c 3 n/ log n a.s. as n → ∞.

Theorem 5 (Recurrent case without local time). Suppose α = d 0 ∈ {1, 2} and Assumption 1. We set

a n := n 2 β (log n) 2-2 β .
Then the finite distributions of ((U ⌊nt⌋ /a n ) t>0 ) n converge to the finite distributions of (K

2 β β Z t,t ) t>0 , with K β := Γ(β + 1)/c β-1 3
and with c 3 given by [START_REF] Cabus | Functional limit theorems for U-statistics indexed by a random walk[END_REF].

When α > d 0 (which implies d 0 = 1), we prove a result of convergence in distribution in the Skorokhod space for the J 1 -metric. Recall that h M (x, y) = h(x, y)1 {|h(x,y)|≤M} + β β-1 (c 0c 1 )M 1-β . Theorem 6 (Recurrent case with local time). Assume α ∈ (1, 2], d 0 = 1 and Assumption 1. We set

a n := n 2δ with δ = 1 -1 α + 1 αβ . Then, for every T > 0, ((U ⌊nt⌋ /a n ) t∈[0,T ] ) n converges in distribution (in the Skorokhod space D([0, T ]) endowed with the J 1 metric) to ( R 2 L t (x)L t (y) dZ x,y ) t∈[0,T ]
, where (L t (x), t ≥ 0, x ∈ R) is a jointly continuous version of the local time at point x at time t of (Y s ) s≥0 (such that, for every t, L t is compactly supported).

Observe that, in every case, there exists c > 0 such that [START_REF] Castell | Limit theorems for one and two-dimensional random walks in random scenery[END_REF]

a n ∼ cn 2 (E[R n ]) 2 β -2
(see for example [23, p. 36] and [19, pp. 698-703]). It is worth noting that U n can be rewritten as follows

U n = x,y∈Z d 0 h(ξ x , ξ y )N n (x)N n (y).
Proposition 7. The results of convergence of finite dimensional distributions of Theorems 3, 5 and 6 hold also if we replace Item (i) of Assumption 1 by the following assumption:

(i') E[exp(iuh(ξ 1 , ξ 1 ))] -1 = O(|u| β ′ ) for some β ′ > β/2.
Observe that (i') includes (i) and the case when h(ξ 1 , ξ 1 ) is in the normal domain of attraction of a β ′ -stable distribution for some β ′ > β/2, in particular this applies if h(ξ 1 , ξ 1 ) has the same distribution as h(ξ 1 , ξ 2 ).

Proof. Due to Theorems 3, 5 and 6, we know that the finite dimensional distributions of

    x =y h(ξ x , ξ y )N ⌊nt⌋ (x)N ⌊nt⌋ (y)/a n   t>0   n converge. It remains to prove that ( x h(ξ x , ξ x )N 2
⌊nt⌋ (x)/a n ) n converges in probability to 0 (for every t > 0). We write ϕ h(ξ1,ξ1) for the characteristic function of h(ξ 1 , ξ 1 ). Let t > 0 and u be two real numbers.

We have

E   exp   iu x∈Z d 0 h(ξ x , ξ x )N 2 ⌊nt⌋ (x) a n     = E   x∈Z d 0 ϕ h(ξ1,ξ1) uN 2 ⌊nt⌋ (x) a n   .
To conclude we just have to prove that

x∈Z d 0 ϕ h(ξ1,ξ1) uN 2 ⌊nt⌋ (x)
an n converges almost surely to 1.

Due to (i'), there exists C 2 > 0 such that we have

x∈Z d 0 ϕ h(ξ1,ξ1) uN 2 ⌊nt⌋ (x) a n -1 ≤ C 2 x∈Z d |u| β ′ N 2β ′ ⌊nT ⌋ (x) a β ′ n
which converges almost surely to 0 since, for every ε > 0, the following inequalities hold almost surely, for n large enough

R n ≤ n 1 α 0 +ε , N * n ≤ n 1-1 α 0 +ε and a -1 n ≤ n -2+ 2 α 0 -2 α 0 β +ε
(see for example [23,[START_REF] Jain | Asymptotic behavior of the local time of a recurrent random walk[END_REF][START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF]).

Examples

The following examples are variants of Example 2.4 from [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF]. Observe that

P(h(ξ 1 , ξ 2 ) > z) = E P(h(x, ξ 2 ) > z) dP ξ1 (x)
and that

P(|h(ξ 1 , ξ 2 )| > z, |h(ξ 1 , ξ 3 )| > z ′ ) = E P(|h(x, ξ 2 )| > z)P(|h(x, ξ 2 )| > z ′ ) dP ξ1 (x).
• When β < 1, one can take E = R p , the distribution of ξ 1 admitting a bounded density f with respect to the Lebesgue measure on E and h(x, y) = xy -p/β ∞ 1 {x =y} . This example fits Assumption 1. Indeed, for every z > 0, P(h(ξ 1 , ξ 2 ) < -z) = 0 and

P(h(x, ξ 2 ) > z) = P( x -ξ 2 ∞ < z -β p ) ∼ z→+∞ 2 p f (x)z -β and P(h(x, ξ 2 ) > z) ≤ f ∞ 2 p z -β . So P(h(ξ 1 , ξ 2 ) > z) ∼ z→+∞ 2 p z -β R d (f (x)) 2 dx
and

P(|h(ξ 1 , ξ 2 )| > z, |h(ξ 1 , ξ 3 )| > z ′ ) ≤ (1 + f ∞ 2 p ) 2 (max(1, z) max(1, z ′ )) -β .
• Analogously, when β ≥ 1, we can take

E = {±1} × R p , h((ε, x), (ε ′ , y)) = εε ′ x -y -p/β ∞
1 {x =y} and ξ 1 = (ε 1 , ξ 1 ) with ε 1 and ξ 1 independent; ε 1 being centered and the distribution of ξ 1 admitting a bounded density f with respect to the Lebesgue measure on R p . Using the same argument as for the previous example together with Remark 2 we can verify that this example satisfies Assumption 1. Note that the case β = 1 contains the more concrete kernel h(x, y) = 1/(x + y) for x = y in association with some random variable ξ 1 having a bounded symmetric density on R.

Convergence of finite distributions

To simplify notations and the presentation of the proofs, we set [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF] |z| β + := |z| β and |z| β -:= |z| β sgn(z) for any real number z. Let m ≥ 1 and θ 1 , ..., θ m ∈ R and 0 = t 0 < t 1 < ... < t m .

If α 0 > 1, we will prove the convergence in distribution of the sequence of random variables ( 9)

  a -1 n x,y∈Z d 0 m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y))h(ξ x , ξ y )   n∈N .
If α 0 = 1, since the limit process will have independent increments, it will be more natural to prove the convergence in distribution of the sequence

  a -1 n x,y∈Z d 0 m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y) -N ⌊nti-1⌋ (x)N ⌊nti-1⌋ (y) h(ξ x , ξ y )   n∈N .
Setting d i,n (x) := N ⌊nti⌋ (x) -N ⌊nti-1⌋ (x), we observe that (10)

m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y) -N ⌊nti-1⌋ (x)N ⌊nti-1⌋ (y) = m i,j=1 θ max(i,j) d i,n (x)d j,n (y)
and hence, if α 0 = 1, it is sufficient to study for fixed θ i,j the sequence of random variables (11)

  a -1 n x,y∈Z d m i,j=1 θ i,j d i,n (x)d j,n (y)h(ξ x , ξ y )   n∈N
(in view of applying the results to the particular case when θ i,j = θ max(i,j) ). Therefore we have to prove the convergence in distribution of (a

-1 n x,y∈Z d 0 χ n,x,y h(ξ x , ξ y )) n , with χ n,x,y := m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y) if α 0 > 1 and χ n,x,y := m i,j=1 θ i,j d i,n (x)d j,n (y) if α 0 = 1.
The basic idea is to identify the sequences in ( 9) and ( 11) as functionals of some sequence of suitably defined point processes and then to use Kallenberg theorem to prove convergence in law of those point processes. More precisely we will define in section 4.2 the sequence of point processes on

R * = R \ {0} defined through N n (ω, ξ) := x,y∈Z d 0 δ a -1 n ζn,x,y(ω)h(ξx,ξy) ,
where (ζ n,x,y ) n,x,y are suitable random variables defined on some suitable probability space ( Ω, F , P) such that, for every integer n, the random variable x,y∈Z d 0 ζ n,x,y h(ξ x , ξ y ) (with respect to P ξ ⊗ P) has the same law as x,y∈Z d 0 χ n,x,y h(ξ x , ξ y ) (with respect to the original probability measure P).

In section 4.1 we prove that the probability space ( Ω, F , P) and the family (ζ n,x,y ) n,x,y can be chosen in such a way to satisfy [START_REF] Dudley | Distances of probability measures and random variables[END_REF] lim

n→+∞ a -β n x,y∈Z d 0 |χ n,x,y | β ± = G± a.s.,
where G is a suitable random variable on ( Ω, F , P). The construction will vary depending on whether

α 0 = 1 or α 0 > 1.
The almost sure convergence in [START_REF] Dudley | Distances of probability measures and random variables[END_REF] will enable us to use Kallenberg theorem in section 4.2 to prove that for almost every ω ∈ Ω the sequence of point processes (N n (ω, .)) n∈N converges in law (with respect to P ξ ) toward a Poisson point process N ω on R * with the following intensity function

z → β|z| -β-1 (c 0 + c 1 ) G+ (ω) + sgn(z)(c 0 -c 1 ) G-(ω) 2 .
In section 4.3 we will see that a

-1 n x,y∈Z d 0 ζ n,x,y (ω)h(ξ x , ξ y ) equals R * w N n (ω, ξ, dw
) which as n goes to infinity converges in distribution toward R * w N ω (dw). We will also see in section 4.3 that this limit follows a stable law with characteristic function Φ (c0+c1) G+ (ω),(c0-c1) G-(ω),β . This will imply the convergence in distribution of the sequences in ( 9) and (11) toward the same stable limit.

4.1.

A result of convergence.

4.1.1. Case α 0 = 1. We define (13) G ± n := a -β n x,y∈Z d 0 m i,j=1 θ i,j d i,n (x)d j,n (y) β ± and G ± := K 2 β m i,j=1 |θ i,j | β ± (t i -t i-1 )(t j -t j-1 ),
where K β is the constant defined in Theorems 3 or 5 (depending on whether the random walk

(S n ) n is transient or recurrent with α = d 0 ). Lemma 8. If α 0 = 1, (G ± n ) n converges
almost surely to G ± . Applying this lemma with θ i,j = θ max(i,j) , we directly obtain the following almost sure equality [START_REF] Franke | Stable limit theorem for U-Statistic processes indexed by a random walk[END_REF]

lim n→∞ a -β n x,y∈Z d 0 m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y) -N ⌊nti-1⌋ (x)N ⌊nti-1⌋ (y) β ± = K 2 β m j=1 |θ j | β ± (t 2 j -t 2 j-1 ).
Proof of Lemma 8. We proceed as in [START_REF] Cerný | Moments and distribution of the local time of a two-dimensional random walk[END_REF][START_REF] Castell | Limit theorems for one and two-dimensional random walks in random scenery[END_REF].

• Let k be a nonnegative integer. Let us prove that

(15) lim n→+∞ (b n,k ) -2 x,y∈Z d 0   m i,j=1 θ i,j d i,n (x)d j,n (y)   k = (K k ) 2 m i,j=1 (θ i,j ) k (t i -t i-1 )(t j -t j-1 ) a.s., with b n,k := n(log n) k-1 if (S n ) n is recurrent (and α = d 0 ) and with b n,k := n if (S n ) n is transient
(extending the definition of K β given in Theorems 3 or 5 to any nonnegative real number β).

Due to [17, p. 10] (transient case) and to [START_REF] Cerný | Moments and distribution of the local time of a two-dimensional random walk[END_REF] (null recurrent case), we know that

(16) ∀i ∈ {1, ..., m}, lim n→∞ (b n,k ) -1 x∈Z d 0 (d i,n (x)) k = K k (t i -t i-1 ) a.s..
Following some argument from [START_REF] Castell | Limit theorems for one and two-dimensional random walks in random scenery[END_REF], we observe that

x,y∈Z d 0   m i,j=1 θ i,j d i,n (x)d j,n (y)   k - x,y∈Z d 0 m i,j=1 (θ i,j ) k (d i,n (x)d j,n (y)) k ≤ max i,j |θ i,j | k ((i1,j1),...,(i k ,j k ))∈I x,y∈Z d 0 k ℓ=1 d i ℓ ,n (x)d j ℓ ,n (y)
≤ max i,j |θ i,j | k    x,y∈Z d 0   m i,j=1 d i,n (x)d j,n (y)   k - x,y∈Z d 0 m i,j=1 (d i,n (x)d j,n (y)) k    ≤ max i,j |θ i,j | k      x∈Z d 0 (N ⌊ntm⌋ (x)) k   2 -   m i=1 x∈Z d 0 (d i,n (x)) k   2    ,
where I denotes the set of ((i 1 , j 1 ), ...,

(i k , j k )) ∈ ({1, ..., m} 2 ) k such that #{(i 1 , j 1 ), ..., (i k , j k )} ≥ 2. Due to (16), we conclude that this term is in o((b n,k ) 2 ). • Assume here that (S n ) n is recurrent and α = d 0 . Let us define W n := (c 3 ) 2 log 2 n m i,j=1 θ i,j d i,n (V n )d j,n (V ′ n ), with (V n , V ′ n ) such that the conditional distribution of (V n , V ′ n )
given S is the uniform distribution on the set {z : N ⌊ntm⌋ (z) ≥ 1} 2 . We observe that ( 17)

E[|W n | u ± |S] = c 2u 3 log 2u n 1 R 2 ⌊ntm⌋ x,y∈Z d 0 m i,j=1 θ i,j d i,n (x)d j,n (y) u ± for all u > 0. Recall that R ⌊ntm⌋ is the cardinal of {z : N ⌊ntm⌋ (z) ≥ 1} and that R n ∼ c 3 n/ log n a.s.. Due to (15) and since K k = Γ(k + 1)/c k-1 3
, we conclude that, for every non negative integer k, we have, almost surely,

lim n→+∞ E[(W n ) k |S] = (Γ(k + 1)) 2 m i,j=1 (θ i,j ) k t i -t i-1 t m t j -t j-1 t m = E[W k ∞ ],
with

W ∞ = θ V,V ′ T T ′ where V ′ , V, T, T ′ are independent random variables, T and T ′ having exponential distribution of parameter 1, V and V ′ being such that P(V = i) = P(V ′ = i) = ti-ti-1 tm
for every i ∈ {1, . . . , m}. From which we conclude that, almost surely, (W n |S) n converges in distribution to W ∞ and that ( 18)

lim n→+∞ E[|W n | β ± |S] = E[|W ∞ | β ± ] a.s..
The proof now follows due to [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF] and [START_REF] Khoshnevisan | Level sets of the stochastic wave equation driven by a symmetric Lévy noise[END_REF].

• Assume now that (S n ) n is transient and set this time

W n := m i,j=1 θ i,j d i,n (V n )d j,n (V ′ n ),
for the same choice of (V n , V ′ n ) as in the previous case. Observe that [START_REF] Guillotin-Plantard | Limit theorems for U-statistics indexed by a one dimensional random walk[END_REF] and since

E[|W n | u ± |S] = 1 R 2 ⌊ntm⌋ x,y∈Z d 0 m i,j=1 θ i,j d i,n (x)d j,n (y) u ± for all u > 0. We recall now, that R n ∼ pn with p := P(S k = 0, ∀k ≥ 1) = 2/(E[N ∞ ] + 1) (see [23, p. 35]). Due to
K k = E[N k-1
∞ ], we obtain that, for every nonnegative integer k, we have almost surely

lim n→+∞ E[W k n |S] = E[N k-1 ∞ ] p 2 m i,j=1 (θ i,j ) k t i -t i-1 t m t j -t j-1 t m . So (W n |S) n converges in distribution to T T ′ θ V,V ′ where V, V ′ , T, T ′ are independent random variables such that ∀i ∈ {1, ..., m}, P(V = i) = P(V ′ = i) = t i -t i-1 t m and ∀m ≥ 1, P(T = m) = P(T ′ = m) = P(N ∞ = m) mp = (1 -p) m-1 p. Indeed, setting N ∞ (0) := sup n N n (0), we have P(N ∞ (0) = k) = (1 -p) k p for every integer k ≥ 0. Note that N ∞ = 1 + N ∞ (0) + Ñ∞ (0)
where Ñ∞ (0) = n≤-1 1 {Sn=0} which is an independent copy of N ∞ (0). Hence we have

P(N ∞ = m) = k,ℓ≥0:k+ℓ=m-1 P(N ∞ (0) = k)P(N ∞ (0) = ℓ) = mp 2 (1 -p) m-1 ,
for every integer m ≥ 1. Therefore

lim n→+∞ E[|W n | β ± |S] = E[N β-1 ∞ ] p 2 m i,j=1 |θ i,j | β ± t i -t i-1 t m t j -t j-1 t m a.s..
This finishes the proof in this case.

Since in the main proof we want to treat simultaneously the cases α 0 = 1 and α 0 > 1, we have to introduce some additional notations which will have its counterpart in the case α 0 > 1. So for α 0 = 1, we set Ñn,ti (x

) := N ⌊nti⌋ (x), Ñ * n := N * ⌊ntm⌋ , Rn := R ⌊ntm⌋ , G± n := G ± n and G± := G ± . We fix ε > 0 such that ε < 1/(3 + 4β) and (3 + 4γ)ε < 4γ β -3. If β < 4/3, we assume moreover that 3 -4 min(1,γ) β + 7ε < 0
(with γ of Item (iv) of Assumption 1). If β ≥ 4/3, we assume that 3 -4(θ ′ +1) β + (4θ ′ + 7)ε < 0 (with θ ′ of Item (vi) of Assumption 1). We write F for the sub-algebra generated by S. We consider the set

Ω0 ∈ F on which (G + n , G - n , n -ε N * n ) converges to (G + , G -, 0)
. When α 0 = 1, we will make no distinction between E and E nor between P and P.

4.1.2. Case α 0 > 1. For every b, t ≥ 0, we set

F n,t (b) := n -1 n 1 α b 0 N ⌊nt⌋ (⌊y⌋) dy, F n,t (-b) := -n -1 0 -n 1 α b N ⌊nt⌋ (⌊y⌋) dy, F t (b) = b 0 L t (x) dx and F t (-b) = - 0 -b L t (x) dx,
(recall that L s (x) is the local time of (Y t ) t at position x and up to time s). It was proved in [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF] that F n,t (b) converges towards F t (b) in distribution. We prove some vector version of this result. Let us define

G ± n := a -β n x,y∈Z m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y) β ± and G ± := R 2 m i=1 θ i L ti (x)L ti (y) β ± dxdy. Lemma 9. The finite distributions of (F n,t1 , • • • , F n,tm , G + n , G - n ) n converge to the finite distributions of (F t1 , • • • , F tm , G + , G -), i.e. ((F n,ti (b j )) i=1,••• ,m,j=1,...,q , G + n , G - n ) n converges in distribution to the random variable ((F ti (b j )) i=1,••• ,m,j=1,...,q , G + , G -) n ,
for every integer q ≥ 1 and every real numbers b 1 , ..., b q .

Proof. The proof of this convergence result follows mainly the proof of Lemma 6 of [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF]. For any real number τ > 0 and any positive integers n and M , we define

V ± (τ, M, n) := τ 2-2β |k|,|ℓ|≤M |T (k, ℓ, n)| β ± ,
where

T (k, ℓ, n) := n -2 m j=1 θ j ⌈(k+1)τ n 1 α ⌉-1 x=⌈kτ n 1 α ⌉ ⌈(ℓ+1)τ n 1 α ⌉-1 y=⌈ℓτ n 1 α ⌉ N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y).
As in [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF], we decompose

G ± n -V ± (τ, M, n) as follows G ± n -V ± (τ, M, n) = U ± (τ, M, n) + W ± 1 (τ, M, n) + W ± 2 (τ, M, n), with U ± (τ, M, n) := n -2δβ (x,y)∈Aτ,M,n m j=1 θ j N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) β ± , where A τ,M,n := Z 2 \ {⌈-M τ n 1 α ⌉, ..., ⌈(M + 1)τ n 1 α ⌉ -1} 2 , W ± 1 (τ, M, n) := |k|,|ℓ|≤M (x,y)∈E k,n ×E ℓ,n n -2δβ W ± 1,k,ℓ (x, y), where E k,n := {⌈kτ n 1 α ⌉, ..., ⌈(k + 1)τ n 1 α ⌉ -1}, W ± 1,k,ℓ (x, y) := m j=1 θ j N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) β ± -n 2β (#E k,n #E ℓ,n ) -β |T (k, ℓ, n)| β ± and W ± 2 (τ, M, n) := |k|,|ℓ|≤M n 2β-2δβ (#E k,n #E ℓ,n ) 1-β -τ 2-2β |T (k, ℓ, n)| β ± .
The proof follows now in five steps:

1) Observe that, due to [17, Lemma 1], there exists a function η satisfying lim x→+∞ η(x) = 0 such that

sup n P U ± (τ, M, n) = 0 ≤ sup n P ∃x : |x| ≥ M τ n 1 α and N ⌊ntm⌋ (x) = 0 = η(M τ ). (19)
2) We prove that there exists some K > 0 and u > 0 such that for all M > 1 one has

sup n E[|W ± 1 (τ, M, n)|] ≤ K(M τ ) 2 τ u . ( 20
)
We first do the case β ≤ 1. Using the fact that

||a| β ± -|b| β ± | ≤ 2 1-β |a -b| β , we have 2 β-1 E[|W ± 1,k,ℓ (x, y)|] ≤ E    m j=1 θ j N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -n 2 (#E k,n #E ℓ,n ) -1 T (k, ℓ, n) β    ≤ m j=1 θ j N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -n 2 (#E k,n #E ℓ,n ) -1 T (k, ℓ, n) β 2 ≤ (#E k,n #E ℓ,n ) -β m j=1 (x ′ ,y ′ )∈E k,n ×E ℓ,n θ j N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -N ⌊ntj ⌋ (x ′ )N ⌊ntj ⌋ (y ′ ) β 2 ≤ (#E k,n #E ℓ,n ) -β 2   m i=1 θ 2 i m j=1 (x ′ ,y ′ )∈E k,n ×E ℓ,n N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -N ⌊ntj ⌋ (x ′ )N ⌊ntj ⌋ (y ′ ) 2 2   β 2
, due to the Cauchy-Schwarz inequality. Now we have to estimate

(x ′ ,y ′ )∈E k,n ×E ℓ,n E |N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -N ⌊ntj ⌋ (x ′ )N ⌊ntj ⌋ (y ′ )| 2 , for (x, y) ∈ E k,n × E ℓ,n . To this end, we use E[|ab -a ′ b ′ | 2 ] ≤ 2 a 2 4 b -b ′ 2 4 + a -a ′ 2 4 b ′ 2 4 together with the fact that sup x E[(N n (x)) 4 ] = O(n 4-4 α ) and sup y =z E[|N n (y) -N n (z)| 4 ] |y -z| 2α-2 = O(n 2-2 α )
(see for example [16, p.77] for the last estimate). This gives, ( 21)

E[|N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -N ⌊ntj ⌋ (x ′ )N ⌊ntj ⌋ (y ′ )| 2 ] ≤ Cτ α-1 n 4-4 α ,
for every (x, y), (x ′ , y ′ ) ∈ E k,n × E ℓ,n and for some C > 0 independent of (τ, M, n, k, ℓ). Therefore, we obtain

E[|W ± 1 (τ, M, n)|] ≤ C ′ (2M + 1) 2 τ 2+ β 2 (α-1) ,
where C ′ does not depend on (τ, M, n). From this we conclude in the case β ≤ 1.

When β > 1, we use ||a| β ± -|b| β ± | ≤ β|a-b|(|a| β-1 + |b| β-1
) combined with the Cauchy-Schwarz inequality and obtain

E[|W ± 1,k,ℓ (x, y)|] ≤ β(#E k,n #E ℓ,n ) -1 m j=1 θ j (x ′ ,y ′ )∈E k,n ×E ℓ,n (N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -N ⌊ntj ⌋ (x ′ )N ⌊ntj ⌋ (y ′ )) 2 × × m j=1 θ j N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) β-1 + n 2 (#E k,n #E ℓ,n ) -1 |T (k, ℓ, n)| β-1 2 ≤ β(#E k,n #E ℓ,n ) -1 m j=1 |θ j | (x ′ ,y ′ )∈E k,n ×E ℓ,n (N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) -N ⌊ntj ⌋ (x ′ )N ⌊ntj ⌋ (y ′ )) 2 × ×    m j=1 θ j N ⌊ntj ⌋ (x)N ⌊ntj ⌋ (y) β-1 2(β-1) + n 2β-2 (#E k,n #E ℓ,n ) 1-β T (k, ℓ, n) β-1 2(β-1)    ≤ C(τ α-1 n 4-4 α ) 1 2 sup x ′ N ⌊ntm⌋ (x ′ ) 2β-2 4(β-1) + +n 2β-2 (τ n 1 α ) 2-2β n -2 (τ n 1 α ) 2 sup x ′ N ⌊ntm⌋ (x ′ ) 2 4(β-1) β-1
, due to the Cauchy Schwarz inequality and to [START_REF] Resnick | Extreme values, regular variation, and point processes[END_REF]. Hence we have

E[|W ± 1,k,ℓ (x, y)|] ≤ C ′ τ α-1 2 n 2-2 α n (1-1 α )2(β-1) = C ′ τ α-1 2 n 2β(1-1 α )
for some C ′ > 0 and so

E[|W ± 1 (τ, M, n)|] ≤ C ′′ (2M + 1) 2 τ 2+ α-1 2
, where C ′′ does not depend on (τ, M, n) and we conclude in the case when β > 1.

3) We notice that

V ± (τ, M ) := τ 2-2β |k|,|ℓ|≤M (k+1)τ kτ (ℓ+1)τ ℓτ m j=1 θ j L tj (x)L tj (y) dxdy β ±
converge to G ± as (τ, M τ ) → (0, ∞), since the local times x → L tj (x) are almost surely continuous and compactly supported (see [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF]). 4) We observe that, for every choice of (τ, M ) the sequence (W ± 2 (τ, M, n)) n converges in probability to 0 as n → ∞. This comes from the fact that for every (k, ℓ) the sequence

(T (k, ℓ, n)) n converges in distribution to m j=1 θ j (k+1)τ kτ L tj (x) dx (ℓ+1)τ ℓτ L tj (y) dy
and the fact that the sequence

(n 2β(1-δ) (#E k,n #E ℓ,n ) 1-β -τ 2-2β
) n converges to 0. 5) For every choice of (τ, M ), for every q and every real numbers b 1 , ..., b q , the sequence of random variables

((F n,ti (b j )) i,j , V + (τ, M, n), V -(τ, M, n))) n converges in distribution to ((F ti (b j )) i,j , V + (τ, M ), V -(τ, M )). Indeed, we recall that V ± (τ, M, n) := τ 2-2β |k|,|ℓ|≤M |T (k, ℓ, n)| β ± and notice that T (k, ℓ, n) =   m j=1 θ j F n,tj ((k + 1)τ ) -F n,tj (kτ ) F n,tj ((ℓ + 1)τ ) -F n,tj (ℓτ )   + O(n -1 )
6) Now we conclude. Let z i,j , z ± ∈ R and ǫ > 0. Due to Points 1, 2 and 3, we fix M > 1 and τ > 0 such, for every n, we have ( 22)

E e i(z+G + n +z-G - n ) -e i(z+(V + (τ,M,n)+W + 2 (τ,M,n))+z-(V -(τ,M,n)+W - 2 (τ,M,n)))
< ε and ( 23)

E e i(z+V + (τ,M)+z-V -(τ,M)) -e i(z+G + +z-G -) < ǫ.
Due to Points 4 and 5 for this choice of (M, τ ), there exists n 0 such that for every n ≥ n 0 , ( 24)

E e iz+W + 2 (τ,M,n)+iz-W - 2 (τ,M,n) -1 < ǫ and (25)
E e i( i,j zij Ft i (bj ))+z+V + (τ,M)+z-V -(τ,M)) -E e i( i,j zij Fn,t i (bj ))+z+V + (τ,M,n)+z-V -(τ,M,n)) < ǫ.
Hence, for every n ≥ n 0 , we have

E e i( i,j zij Ft i (bj ))+z+G + +z-G -) -E e i( i,j zij Fn,t i (bj ))+z+G + n +z-G - n ) ≤ 3ǫ + E e i( i,j zij Ft i (bj ))+z+V + (τ,M)+z-V -(τ,M)) -E e i( i,j zij Fn,t i (bj ))+z+V + (τ,M,n)+z-V -(τ,M,n))
≤ 4ε.

where we used ( 22), ( 23), (24) for the first inequality and (25) for the last one.

Let C be the set of continuous functions g : R → [-t m , t m ]. We endow this set with the following metric D corresponding to the uniform convergence on every compact:

D(g, h) := N ≥1 2 -N sup x∈[-N ;N ] |g(x) -h(x)|. Lemma 10. The sequence (F n,t1 , ..., F n,tm ) n∈N is tight in (C, D) m .
Proof. It is enough to prove the tightness of F n,ti for all i ∈ {1, ..., m}. To simplify notations in this proof we use F n to denote F n,ti /t i and F to denote F ti /t i . As usual, for any f ∈ C, we denote by ω(f, •) the modulus of continuity of f . Since F n (0) = 0 for every n, it is enough to prove

(26) ∀ε > 0, lim δ→0 lim sup n→+∞ P(ω(F n , δ) ≥ ε) = 0 (see [2, p.83]). Let ε > 0 and ε 0 > 0. Let M > 0 be such that P |F (M ) -F (-M )| ≤ 1 -(ε/2) ≤ ε 0 /2. Since (F n (M ) -F n (-M )) n converges in distribution to F (M ) -F (-M ), we have (27) lim sup n→+∞ P |F n (M ) -F n (-M )| ≤ 1 -(ε/2) ≤ P |F (M ) -F (-M )| ≤ 1 -(ε/2) ≤ ε 0 /2.
Let δ 0 > 0 be such that, for every δ ∈ (0, δ 0 ), P(ω(F, δ) ≥ ε/2) ≤ ε 0 /2 (since F is almost surely uniformly continuous). Since the finite distributions of (F n ) n converge to the finite distribution of F , we have

lim sup n→+∞ P ∃k = - M δ , ..., M δ , |F n (kδ) -F n ((k + 1)δ)| ≥ ε 2 (28) ≤ P ∃k = - M δ , ..., M δ , |F (kδ) -F ((k + 1)δ)| ≥ ε 2 ≤ P ω(F, δ) ≥ ε 2 ≤ ε 0 2 .
Putting ( 27) and ( 28) together, we obtain that, for every δ < δ 0 , we have

lim sup n→+∞ P(ω(F n , δ) ≥ ε) ≤ lim sup n→+∞ P ∃k = - M δ , ..., M δ , |F n (kδ) -F n ((k + 1)δ)| ≥ ε 2 + lim sup n→+∞ P |F n (M ) -F n (-M )| ≤ 1 -(ε/2)
and so lim sup

n→+∞ P(ω(F n , δ) ≥ ε) ≤ ε 0 .
Due to Lemma 9 and Lemma 10, the sequence

(F n,t1 , . . . , F n,tm , G + n , G - n ) n converges in distribution to (F t1 , . . . , F tm , G + , G -) in (C, D) m × (R, | • |) 2 . We fix ε ∈ (0, βδ/(1 + β)) such that (3 + 4β)ε < 1/α and (3 + 4γ)εα < 4γ
β -3 (this is possible due to γ > 3β/4). If β < 4/3, we assume moreover that 3 α -4 min(1,γ) αβ + 7ε < 0 (with γ of Item (iv) of Assumption 1). If β ≥ 4/3, we assume also that 1 α 3 -4(θ ′ +1) β + (4θ ′ + 7)ε < 0 (with θ ′ of Item (vi) of Assumption 1). Using for example [START_REF] Jain | Asymptotic behavior of the local time of a recurrent random walk[END_REF] for the maximal occupation time and appendix of [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF] for the range, we know that (n -1/α-ε R n , n (1/α)-1-ε N * n ) n converges almost surely to 0. Therefore the sequence (F n,t1 , . . . , 4 . Now using the Skorokhod representation theorem (see [START_REF] Dudley | Distances of probability measures and random variables[END_REF] p.1569) (since (C, D) and R are separable and complete), we know that there exists a probability space ( Ω, F , P) with random variables ( Fn,t1 , . . . , Fn,tm , G+ n , Gn , Rn , Ñ * n ) n and ( Ft1 , . . . , Ftm , G+ , G-) defined on ( Ω, F , P) such that

F n,tm , G + n , G - n , n -1/α-ε R n , n (1/α)-1-ε N * n ) n converges in distribution to (F t1 , . . . , F tm , G + , G -, 0, 0) in (C, D) m × (R, | • |)
• for every integer n, ( Fn,t1 , . . . , Fn,tm , G+ n , Gn , Rn , Ñ * n ) has the same distribution (with respect to

P) as (F n,t1 , . . . , F n,tm , G + n , G - n , R ⌊ntm⌋ , N * ⌊ntm⌋ ) (with respect to P) in (C, D) m × (R, | • |) 4 ; • ( Ft1 , . . . , Ftm , G+ , G-) has the same distribution as (F t1 , . . . , F tm , G + , G -) in (C, D) m × (R, | • |) 4 ; • the sequence ( Fn,t1 , . . . , Fn,tm , G+ n , G- n , n -1/α-ε Rn , n (1/α)-1-ε Ñ * n ) n converges almost surely to ( Ft1 , . . . , Ftm , G+ , G-, 0, 0) in (C, D) m × (R, | • |) 4 .
Observe that, for every x ∈ Z and every n ≥ 1,

N n (x) : f → n(f ((x + 1)n -1 α ) -f (xn -1 α )
) is a continuous functional of (C, D) and that N ⌊nti⌋ (x) = N n (x)(F n,ti ) (for every i ∈ {1, . . . , m}). Therefore, for every integers x and n ≥ 1, for every i ∈ {1, . . . , m}, we define Ñn,ti (x) := N n (x)( Fn,ti ).

Observe that, for every integer N ≥ 1, ( Ñn,ti (x)) x∈{-N,...,N };i∈{1,...,m} , Ñ * n , Rn , G± n has the same distribution as

(N ⌊nti⌋ (x)) x∈{-N,...,N };i∈{1,...,m} , N * ⌊ntm⌋ , R ⌊ntm⌋ , G ± n .
In particular Ñn,ti (x) takes integer values and 0 ≤ Ñn,ti (x) ≤ Ñn,tm (x). Moreover we have the following result.

Lemma 11. Let n be a positive integer. We have Proof. (29) comes from the fact that, for every integers x and n ≥ 1, Ñ * n -Ñn,tm (x) has the same distribution as N * ⌊ntm⌋ -N ⌊ntm⌋ (x) which is non negative. To prove (30), we observe that

(29) sup x∈Z Ñn,tm (x) ≤ Ñ * n , (30)
Rn -#{x ∈ Z : Ñn,tm (x) > 0} = lim N →+∞ Rn -# x ∈ {-N, . . . , N } : Ñn,tm (x) > 0 .
But, for every N ≥ 1, Rn -# x ∈ {-N, . . . , N } : Ñn,tm (x) > 0 has the same distribution as R ⌊ntm⌋ -# x ∈ {-N, . . . , N } : N ⌊ntm⌋ (x) > 0 which converges to 0 as N goes to infinity. This gives (30) by uniqueness of the limit for the convergence in probability.

Finally, we observe that G± n -n -2βδ

x,y∈Z m i=1 θ i Ñn,ti (x) Ñn,ti (y)

G ± n -n -2βδ |x|,|y|≤N m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y) β ± .
But this last random variable converges to 0 as N goes to infinity and we obtain (31).

Let us write (Ω, F , P) for the original space on which ξ and S are defined. We denote F ξ for the sub-σ-algebra of F generated by ξ and P ξ for the restriction of P to F ξ . Now we define (Ω, T , P) as the direct product of (Ω, F ξ , P ξ ) with ( Ω, F , P). We observe that P ξ (•) = P(•| F ).

Lemma 12. For every integer n ≥ 1, the random variable Ãn := x,y∈Z m i=1 θ i Ñn,ti (x) Ñn,ti (y)h(ξ x , ξ y ) has the same distribution (with respect to P) as A n := x,y∈Z m i=1 θ i N ⌊nti⌋ (x)N ⌊nti ⌋ (y)h(ξ x , ξ y ) (with respect to P).

Proof. We proceed as in the proof of Lemma 11. Observe that Ãn is the limit as N goes to infinity of Ãn,N := |x|,|y|≤N m i=1 θ i Ñn,ti (x) Ñn,ti (y)h(ξ x , ξ y ) which has the same distribution as A n,N := |x|,|y|≤N m i=1 θ i N ⌊nti⌋ (x)N ⌊nti⌋ (y)h(ξ x , ξ y ). But A n = lim N →+∞ A n,N . We conclude by unicity of the limit for the convergence in distribution.

Let Ω0 ⊂ Ω be the set of P-measure one on which ( Fn,t1 , . . . , Fn,tm

, G+ n , G- n , n -1/α-ε Rn , n (1/α)-1-ε Ñ * n ) n converges to ( Ft1 , . . . , Ftm , G+ , G-, 0, 0) in C m × R 4 .
4.2. A conditional limit theorem for some associated point process. To simplify notations, we set (32)

ζ n,x,y := m i=1 θ i Ñn,ti (x) Ñn,ti (y) if α 0 > 1 and (33) ζ n,x,y := m i,j=1 θ i,j d i,n (x)d j,n (y) if α 0 = 1.
With these notations we have

G± n = a -β n x,y |ζ n,x,y | β ± .
For every ω ∈ Ω0 , we consider the point process N n on R * defined by

N n (ω, ξ)(dz) := x,y∈Z:x =y δ a -1 n ζn,x,y(ω)h(ξx,ξy) (dz).
We already mentioned in ( 7) that

a n ∼ cn 2 (E[R n ]) 2
β -2 for some c > 0 and observe that in any case (34)

∀γ 0 > 0, a -1 n = o n -2+ 2 α 0 -2 α 0 β +γ0 .
Moreover note that for the ǫ > 0 which was fixed in the previous subsection we have

n 1 α 0 -1-ǫ Ñ * n a.s.
-→ 0 and n

-1 α 0 -ǫ Rn a.s.
-→ 0.

In the following we will prove that the sequence of point processes N n ; n ∈ N converges toward some Poisson point process for P almost all ω ∈ Ω. We will essentially follow the notation from [START_REF] Resnick | Extreme values, regular variation, and point processes[END_REF] and denote by M p (R * ) the set of point measures on R * . Further, M p (R * ) is the smallest σ-algebra containing all sets A of the form

A = {m ∈ M p (R * ); m(F ) ∈ B} for some F ∈ B(R * ) and B ∈ B([0, ∞]). We introduce the following metric on R * d(x, y) := | log(x/y)| if sgn(x) = sgn(y); | log |x|| + | log |y|| + 1 if sgn(x) = sgn(y).
With this metric R * becomes a complete separable metric space. We will denote by C K (R * ) the space of continuous functions f : R * → R with compact support with respect to this metric. A sequence of Radon measures µ n is said to converge with respect to the vague topology toward some Radon measure µ if for all f ∈ C K (R * ) one has

lim n→∞ R * f dµ n = R * f dµ.
It is well known that the vague topology on the Radon measures can be generated by some metric which turns it into a complete metric space (see [START_REF] Resnick | Extreme values, regular variation, and point processes[END_REF] p.147) and that the set of point measures is closed in the vague topology (see [START_REF] Resnick | Extreme values, regular variation, and point processes[END_REF] p.145). We will say that a sequence of point processes N n ; n ∈ N converges in distribution toward a Point process N if for all bounded vaguely continuous functions

F : M p (R * ) → R we have lim n→∞ E[F (N n )] = E[F (N )].
Proposition 13. For every ω ∈ Ω0 , N n (ω, •) converges in distribution (with respect to P ξ ) to a Poisson process N ω on R \ {0} of intensity η ω given by

η ω ([d, d ′ )) = (d -β -d ′ -β ) (c 0 + c 1 ) G+ (ω) + (c 0 -c 1 ) G-(ω) 2 ,
and

η ω ((-d ′ , -d]) = (d -β -d ′ -β ) (c 0 + c 1 ) G+ (ω) -(c 0 -c 1 ) G-(ω) 2 ,
(with convention ∞ -β = 0) for every 0 < d < d ′ ≤ +∞.

Proof. Our proof is based on some method presented in [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF]. Due to Kallenberg's theorem [START_REF] Resnick | Extreme values, regular variation, and point processes[END_REF], it is enough to prove that, for any finite union R = K i=1 Q i of intervals, where

Q i := [d i , d ′ i ) ⊂ (0, +∞) or Q i = (-d ′ i , -d i ] ⊂ (-∞, 0). We have (35) lim n→+∞ E[N n (R)| F ](ω) = η ω (R)
and ( 36)

lim n→+∞ P(N n (R) = 0| F )(ω) = e -ηω(R) .
We start with the proof of (35). By linearity, it is enough to prove it for a single interval Q. For any interval Q = [d, d ′ ) ⊂ (0, +∞), since ξ is a sequence of iid random variables, we have

E[N n (Q)| F ] =
x,y∈Z d 0 :x =y

P(A n,x,y | F )1 {ζn,x,y>0} + P(B n,x,y | F )1 {ζn,x,y<0} , with A n,x,y := a n d|ζ n,x,y | -1 ≤ h(ξ 1 , ξ 2 ) < a n d ′ |ζ n,x,y | -1 and B n,x,y := a n d|ζ n,x,y | -1 ≤ -h(ξ 1 , ξ 2 ) < a n d ′ |ζ n,x,y | -1 .
Observe that, due to (34) and to

Ñ * n = o(n 1-1 α 0 +ε ), we have (37) ∀γ 0 > 0, a -1 n sup x,y |ζ n,x,y | ≤ Ca -1 n ( Ñ * n ) 2 ≤ n -2 α 0 β +2ε+γ0 ,
for n large enough (and for some constant C > 0 depending on θ i or on θ i,j ). Now, combining this with Item (iii) of Assumption 1, we have

x,y:x =y

P(A n,x,y | F )1 {ζn,x,y>0} = c 0 (d -β -d ′ -β )a -β n x,y∈Z d 0 :x =y |ζ n,x,y | β sgn(ζ n,x,y ) + 1 2 ×   1 + O   sup z>n 2 α 0 β -2ε-γ 0 |L 0 (z) -c 0 |     + o(1) = c 0 (d -β -d ′ -β ) G+ n + G- n 2 + o(1),
since ε < 1/(α 0 β) and since, for n large enough,

x∈Z d 0 |ζ n,x,y | β ≤ n 1 α 0 +ε n 2β-2β α 0 +2εβ = o(a β n ),
since ε < 1/((1 + 2β)α 0 ). Analogously, we have

x,y:x =y

P(B n,x,y | F )1 {ζn,x,y<0} = c 1 (d -β -d ′ -β )a -β n x,y∈Z d 0 :x =y |ζ n,x,y | β 1 -sgn(ζ n,x,y ) 2 ×   1 + O   sup z>n 2 α 0 β -2ε-γ 0 |L 1 (z) -c 1 |     + o(1) = c 1 (d -β -d ′ -β ) G+ n -G- n 2 + o(1),
We obtain (35) for Q = [d, d ′ ) ⊂ (0, +∞) using ( 1), ( 2) and the definition of G± n and of G± . The proof of (35) for Q = (-d ′ , -d] ⊂ (-∞, 0) follows the same scheme. Now let us prove (36). Let K ≥ 1 and let R be a union of

K pairwise disjoint intervals Q 1 , ..., Q K with Q i := (d i , d ′ i ] ⊂ (0, +∞) or Q i := [-d ′ i , -d i ) ⊂ (-∞, 0). We write P ω n for the Poisson distribution of intensity η ω n (R) := E[N n (R)| F ](ω). On Ω0 , due to (35), we have |e -ηω(R) -P ω n (0)| = o(1)
. Hence, to prove (36), we just have to prove

(38) |P(N n (R) = 0| F ) -P n (0)| = o(1).
Following [START_REF] Barbour | Poisson convergence for dissociated statistics[END_REF] and [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF], we introduce the following notations. For every x, y ∈ Z d0 such that x = y, we define the random variables

I x,y = K i=1
1 {h(ξx,ξy)∈an(ζn,x,y) -1 Qi} .

Observe that (39) N n (R) =

x,y∈Z d 0 :x =y I x,y and so η n (R) =

x,y∈Z d 0 :x =y

E[I x,y | F ].
We will use the following lemma, whose proof is postponed until the end of this paragraph:

Lemma 14. We have

|P(N n (R) = 0| F ) -P n (0)| ≤ min(1, (η n (R)) -1 )(A 1 + A 2 ),
with

A 1 := (x,y)∈M E[I x,y | F ]E    I x,y + (x ′ ,y ′ )∈M (1)
x,y

I x ′ ,y ′ F    , A 2 := (x,y)∈M E    I x,y   Ix,y + (x ′ ,y ′ )∈M (1)
x,y

I x ′ ,y ′    F    ,
and with the notation M (k)

x,y := {(x ′ , y ′ ) ∈ M : #{x ′ , y ′ } ∩ {x, y} = k} and M := {(x, y) ∈ Z 2d0 : x = y}.

To conclude, we have to prove that A 1 and A 2 converge to 0 as n goes to infinity. We set d := min i d i .

For A 1 , using ( 1), ( 2) and the definition of I x,y , we observe that, for γ 0 > 0 small enough, we have 34) together with the definitions of Rn and Ñ * n (with C some constant depending on θ j and θ i,j ). Now let us study A 2 . We have, for γ 0 > 0 small enough,

A 1 ≤ 4 x,y∈Z d 0 x ′ ∈Z d 0 P da n |ζ n,x,y | -1 ≤ |h(ξ x , ξ y )| F ×P da n |ζ n,x,x ′ | -1 ≤ |h(ξ x , ξ x ′ )| F ≤ Cd -2β a -2β n ( L 0 ∞ + L 1 ∞ ) 2 R3 n ( Ñ * n ) 4β ≤ O(n -1 α 0 +(4β+3)ε+γ0 ) = o(1), using ε(4β +3) < 1/α 0 , (
A 2 ≤ 4 x,y,x ′ ∈Z d 0 P da n |ζ n,x,y | -1 ≤ |h(ξ x , ξ y )|, da n |ζ n,x,x ′ | -1 ≤ |h(ξ x , ξ x ′ )| F ≤ 4C 0 R3 n a -2γ n ( Ñ * n ) 4γ ≤ O n 3 α 0 +(3+4γ)ε-4γ α 0 β +γ0 = o(1), due to (3 + 4γ)εα 0 < 4γ β -3 (recall that this is possible since γ > 3β/4
) and where C 0 is a constant depending on on d, θ j and θ i,j .

Proof of Lemma 14. The proof of this lemma follows the line of arguments that can be found in [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF]. Let f be defined on N by f (0) = 0 and f (m) := e ηn(R) (m -1)! (η n (R)) m P n ({0})P n ([m, +∞)).

We will use the two following inequalities (see [START_REF] Barbour | Poisson convergence for dissociated statistics[END_REF] p.400 and p.401) (40)

P(N n (R) = 0| F ) -P n (0) ≤ E η n (R)f (N n (R) + 1) -N n (R)f (N n (R)) F and (41) sup m |f (m + 1) -f (m)| ≤ min(1, (η n (R)) -1
).

Now we observe that, for every (x, y) ∈ (Z d0 ) 2 such that x = y, we have

(42) N n (R) = x ′ ,y ′ ∈Z d 0 :x ′ =y ′ I x ′ ,y ′ = I x,y + N (0) n,x,y + N (1) n,x,y , with N (i) n,x,y := (x ′ ,y ′ )∈M (i)
x,y I x ′ ,y ′ . Starting from (40) and using (39), we have

P(N n (R) = 0| F ) -P n (0) ≤ A ′ 1 + A ′ 2 , with A ′ 1 := x,y∈Z d 0 :x =y E[I x,y | F ]E f (N n (R) + 1) -f (N (0) n,x,y + 1) F and A ′ 2 := x,y∈Z d 0 :x =y E I x,y f (N n (R)) F -E[I x,y | F ]E f (N (0) n,x,y + 1) F .
Now, using (41) and (42), we obtain

f (N n (R) + 1) -f (N (0) n,x,y + 1) ≤ sup m≥0 |f (m + 1) -f (m)| × N n (R) -N (0) n,x,y ≤ min(1, (η n (R)) -1 )(I x,y + N (1) n,x,y ) (43) and so A ′ 1 ≤ min(1, (η n (R)) -1 )A 1 .
Observe that, conditioned with respect to F , I x,y and N (0) n,x,y are independent. Therefore

A ′ 2 =
x,y∈Z d 0 :x =y

E I x,y {f (N n (R)) -f (N (0) n,x,y + 1)} F .
Now, using (41) once again, we obtain

f (N n (R)) -f (N (0) n,x,y + 1) ≤ min(1, (η n (R)) -1 )(N n (R) -N (0) n,x,y ) ≤ min(1, (η n (R)) -1 )(I x,y + N (1)
n,x,y )

and so A ′ 2 ≤ min(1, (η n (R)) -1 )A 2 , which completes the proof of the lemma. 4.3. Proof of the convergence of the finite dimensional distributions. In this paragraph we will finish the proof of the convergence of the finite dimensional distributions. Similarly to the proof given in [START_REF] Dabrowski | Poisson limits for U-statistics[END_REF], we will use the convergence of the associated point process and the continuous mapping theorem. The approach is based on the following observation:

a -1 n x,y ζ n,x,y h(ξ x , ξ y ) = R * w dN n (w).
However the functional is not continuous and we will have to do some truncation. This will be the purpose of the three following propositions.

Proposition 15. Let δ > 0. For P almost every ω ∈ Ω0 , the sequence of random variables

Z ω n := a -1 n x,y ζ n,x,y (ω)h(ξ x , ξ y )1 {a -1 n |ζn,x,y(ω)h(ξx,ξy)|>δ} = R * w1 (δ,+∞) (|w|) dN ω n (w)
converges in distribution to R * w1 (δ,+∞) (|w|) dN ω (w).

Proposition 16. For every γ 0 > 0, we have

lim δ→0 lim sup n→∞ P |T n (δ)| > γ 0 | F = 0 P -a.s., with T n (δ) := a -1 n x,y ζ n,x,y h(ξ x , ξ y )1 {a -1 n |ζn,x,yh(ξx,ξy)|≤δ} if β ≤ 1 and T n (δ) := a -1 n x,y ζ n,x,y h(ξ x , ξ y )1 {a -1 n |ζn,x,yh(ξx,ξy)|≤δ} + (c 0 -c 1 ) βδ 1-β β -1 G- n if β > 1.
Proposition 17 (see [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes[END_REF]). Let P be a Poisson process on R * with intensity admitting the density z → β|z| -β-1 (a1 {z>0} + b1 {z<0} ).

If β < 1, then R * \[-δ,δ]
w dP(w) converges in distribution, as δ goes to 0, to a stable random variable with characteristic function Φ a+b,a-b,β with the notation of [START_REF] Borodin | A limit theorem for sums of independent random variables defined on a recurrent random walk[END_REF].

If β = 1, then R * \[-δ,δ] w dP(w) -(a -b) +∞ δ sin x
x 2 dx converges in distribution, as δ goes to 0, to a stable random variable with characteristic function Φ a+b,a-b,1 , with the notation of [START_REF] Borodin | Limit theorems for sums of independent random variables defined on a transient random walk[END_REF].

If β > 1, then R * \[-δ,δ] w dP(w) -(a -b) βδ 1-β
β-1 converges in distribution, as δ goes to 0, to a stable random variable with characteristic function Φ a+b,a-b,β with the notation of ( 4).

The following corollary is a consequence of Propositions 13, 15, 16 and 17. Corollary 18. We have Proof of Corollary 18. Observe first that due to the Lebesgue dominated convergence theorem it is enough to prove the first convergence. Let Ω1 be the subset of Ω0 on which the convergences of Propositions 15 and 16 hold and let ω ∈ Ω1 . To simplify notations, let us write

lim n→+∞ E[e ia
V n := a -1 n x,y ζ n,x,y h(ξ x , ξ y ) and W n (δ) := a -1 n x,y ζ n,x,y h(ξ x , ξ y )1 {a -1 n |ζn,x,yh(ξx,ξy)|>δ} .
We set κ := 0 if β ≤ 1 and κ := (c 0c 1 ) β β-1 if β > 1 (recall that we assume c 0 = c 1 if β = 1). We also write W ω (δ) := R\[-δ,δ] w dN ω (w) (where N ω is the Poisson process of Proposition 13, which is defined on some probability space (Ω ω , T ω , P ω) endowed with the expectation E ω ). Let ǫ > 0. Due to Propositions 16, 13 and 17, we consider δ > 0 and n 0 such that, for every n ≥ n 0 , we have (44)

P |T n (δ)| > ǫ 6 F (ω) < ǫ 6 and such that (45) E ω e i(Wω (δ)-κδ 1-β G-(ω)) -Φ (c0+c1) G+ (ω),(c0-c1) G-(ω),β (1)
< ǫ 6 .

Due to Proposition 15, we consider n 1 ≥ n 0 such that, for every n ≥ n 1 , we have

(46) E[e iWn(δ) | F ](ω) -E ω[e iWω (δ) ] < ǫ 6 .
Now, let n 2 ≥ n 1 such that, for every n ≥ n 2 , we have

(47) e iκδ 1-β G-(ω) -e iκδ 1-β G- n (ω) < ǫ 6 .
For n ≥ n 2 , we have Wω (δ) ] due to (47)

E[e iVn | F ](ω) -Φ (c0+c1) G+ (ω),(c0-c1) G-(ω),β (1)
≤ ǫ 6 + E[e iVn | F ](ω) -E ω[e i(Wω (δ)-κδ 1-β G-(ω)) ] due to (45) ≤ ǫ 6 + E[e i(Vn+κδ 1-β G-) | F ](ω) -E ω [e i(Wω (δ) ] ≤ 2ǫ 6 + E[e i(Vn+κδ 1-β G- n ) | F ](ω) -E ω[e i(
≤ 2ǫ 6 + E[e i(Wn(δ)+Tn(δ)) | F ](ω) -E ω [e i(Wω (δ) ] ≤ 3ǫ 6 + E[e i(Wn(δ)+Tn(δ)) -e iWn(δ) | F ](ω) due to (46) ≤ 4ǫ 6 + 2P |T n (δ)| > ǫ 6 F (ω) ≤ ǫ due to (44).
Proof of the convergence of finite distributions in Theorems 3, 5 and 6. Admitting Propositions 15, 16 and 17 for the moment, let us end the proof of the convergence of the finite distributions. Due to Corollary 18, we have

lim n→+∞ E[e ia -1 n x,y ζn,x,yh(ξx,ξy) ] = E Φ (c0+c1) G+ ,(c0-c1) G-,β (1)
= E exp - +∞ 0 sin t t β dt (c 0 + c 1 )G + -i(c 0 -c 1 )G -tan πβ 2 .
When α 0 = 1, with the use of ( 10) and ( 14) , we obtain

lim n→+∞ E e ia -1 n m j=1 θj(U ⌊nt j ⌋ -U ⌊nt j-1 ⌋ ) = exp -K 2 β m i=1 (t 2 i -t 2 i-1 )|θ i | β +∞ 0 sin t t β dt (c 0 + c 1 ) -i(c 0 -c 1 ) sgn(θ i ) tan πβ 2 = m j=1 Φ (c0+c1)K 2 β (t 2 i -t 2 i-1 ),(c0-c1)K 2 β (t 2 i -t 2 i-1 ),β (θ j )
This gives the convergence of the finite distributions in Theorems 3 and 5. When α 0 > 1, due to Lemma 12, we obtain

(48) lim n→+∞ E e i m j=1 θja -1 n U ⌊nt j ⌋ = E Φ (c0+c1)G + ,(c0-c1)G -,β (1)
,

with G ± = R 2 | m i=1 θ i L ti (x)L ti (y)| β ± dxdy.
Let us recall that the right hand side of (48) corresponds to the characteristic function of m i=1 θ i R 2 L ti (x)L ti (y) dZ x,y evaluated at one (see for example [START_REF] Khoshnevisan | Level sets of the stochastic wave equation driven by a symmetric Lévy noise[END_REF] and Appendix A).

Proof of Proposition 15.

To simplify notations we also write P ω for P(•| F )(ω) and

E ω for E[•| F ](ω).
We proceed in four steps: 1) We first use the continuous mapping theorem (see [START_REF] Resnick | Extreme values, regular variation, and point processes[END_REF] p.151) to prove that for P-almost all ω one has

(-M,-δ)∪(δ,M) zdN ω n (dz) L -→ (-M,-δ)∪(δ,M) zdN ω (dz). (49)
The Poisson process Ñω has P-almost surely only a finite number of points in the interval (-M, -δ) ∪ (δ, M ). Moreover, one has P-almost surely that each of those points only carries the mass one, since the Poisson process Ñω is simple. Now, let µ be a point measure with only a finite number of points with mass one in (-M, -δ) ∪ (δ, M ) and let (µ n ) n∈N be some sequence of point measures which converges toward µ with respect to the vague topology on R * . Let {x 1 , ..., x p } be the support of µ intersected with (-M, -δ) ∪ (δ, M ). According to [START_REF] Neveu | Processus ponctuels, Ecole d'été de Probabilité de Saint-Flour[END_REF] (see Lemma I.14) there exists some large N ∈ N such that for all n ≥ N the support of µ n intersected with (-M, -δ) ∪ (δ, M ) in exactly p point x

(n) 1 , ..., x (n) p such that lim n→∞ x (n) i = x i for all i = 1, ..., p.

It then follows that

lim n→∞ (-M,-δ)∪(δ,M) zµ n (dz) = lim n→∞ p i=1 x (n) i = p i=1 x i = (-M,-δ)∪(δ,M) zµ(dz).
2) We now prove that for P-almost all ω one has

(-∞,-M)∪(M,∞) zdN ω (dz) Pω -→ 0 as M → ∞. ( 50
)
This follows from the following equality which holds for P-almost all ω

E ω exp it ∞ M zN ω (dz) = exp (c 0 + c 1 ) G+ ∞ M β cos(tx) -1 x β+1 dx + i(c 0 -c 1 ) G- ∞ M β sin(tx) x β+1 dx
and from the fact that one has

(c 0 + c 1 ) G+ ∞ M β cos(tx) -1 x β+1 dx + i(c 0 -c 1 ) G- ∞ M β sin(tx) x β+1 dx ≤ 2M -β (c 0 + c 1 )(| G+ | + | G-|) .
This yields

E ω exp it ∞ M zN ω (dz) -→ 1 for P almost all ω as M → ∞.
The convergence in probability follows from the convergence in law of ∞ M zN ω (dz) toward zero. The other part -M -∞ zN ω (dz) is treated in the same way.

3) We now prove that for P-almost all ω we have

sup n∈N P ω (-∞,-M)∪(M,∞) zN ω n (dz) = 0 -→ 0 as M → ∞. (51)
For this first remember that

(-∞,-M)∪(M,∞) zN ω n (dz) = x,y∈Z a -1 n ζ n,x,y h(ξ x , ξ y )1 {|a -1 n ζn,x, yh(ξx,ξy)|>M} .
Thus this implies

P ω {|z|>M} zN ω n (dz) = 0 ≤ P ω ∃x, y ∈ Z : |a -1 n ζ n,x,y h(ξ x , ξ y )| > M ≤ x,y∈Z P ω |h(ξ x , ξ y )| > M a n |ζ n,x,y | -1 ≤ x,y∈Z C M a n |ζ n,x,y | -1 -β ≤ CM -β a -β n x,y∈Z |ζ n,x,y | β = CM -β G + n -→ 0 as M → ∞, since P-almost surely we have G + n → G + as n → ∞. 4)
We now use the previous findings to conclude. We consider an ω which satisfies all the requirements from points (1) to (3) of this proof. For some given t ∈ R and ǫ > 0 we use (51) to find some M > 0 such that

sup n∈N P ω (-∞,-M)∪(M,∞) zN ω n (dz) = 0 ≤ ǫ/8
By (50) we can assume without loss of generality that the M also satisfies

P ω t (-∞,-M)∪(M,∞) zdN ω (dz) ≥ ǫ/4 ≤ ǫ/8.
Moreover, according to (49) we can find some n 0 ∈ N such that for all n ≥ n 0 we have

E ω exp it (-M,-δ)∪(δ,M) zN ω n (dz) -E ω exp it (-M,-δ)∪(δ,M)
zN ω (dz) ≤ ǫ/4.

It now follows that

E ω exp it (-∞,-δ)∪(δ,∞) zN ω n (dz) -E ω exp it (-∞,-δ)∪(δ,∞) zN ω (dz) = E ω exp it (-M,-δ)∪(δ,M) zN ω n (dz) 1 + exp it (-∞,-M)∪(M,∞) zN ω n (dz) -1 -E ω exp it (-M,-δ)∪(δ,M) zN ω (dz) 1 + exp it (-∞,-M)∪(M,∞) zN ω (dz) -1 ≤ E ω exp it (-M,-δ)∪(δ,M) zN ω n (dz) -E ω exp it (-M,-δ)∪(δ,M) zN ω (dz) +2P ω (-∞,-M)∪(M,∞) zN ω n (dz) = 0 + 2P ω t (-∞,-M)∪(M,∞) zdN ω (dz) ≥ ǫ/4 + ǫ 4 .
Since the right side is equal to ǫ this finishes the proof of the proposition.

Proof of Proposition 16.

• When β < 1, we just prove that lim δ→0 lim sup n→∞ E[|T n (δ)|| F ] = 0. Due to Item (iii) of Assumption 1, we have

E[|T n (δ)|| F ] ≤ x,y E a -1 n |ζ n,x,y h(ξ x , ξ y )|1 {a -1 n |ζn,x,yh(ξx,ξy)|≤δ} F ≤ x,y δ 0 P δ ≥ a -1 n |h(ξ x , ξ y )ζ n,x,y | > z F dz ≤ x,y δ 0 P |h(ξ x , ξ y )ζ n,x,y | > a n z F dz ≤ ( L 0 ∞ + L 1 ∞ ) x,y δ 0 a -β n z -β (ζ n,x,y ) β dz ≤ ( L 0 ∞ + L 1 ∞ ) x,y a -β n δ 1-β 1 -β (ζ n,x,y ) β ≤ ( L 0 ∞ + L 1 ∞ ) δ 1-β 1 -β G+ n . So lim δ→0 lim sup n→∞ E[|T n (δ)|| F ] ≤ lim δ→0 ( L 0 ∞ + L 1 ∞ ) δ 1-β 1-β G+ = 0, since β < 1.
• Assume here that β ∈ (1, 2). Observe that, due to Item (v) of Assumption 1, we have

E h(ξ 1 , ξ 2 )1 {|h(ξ1,ξ2)|≤M} = -E h(ξ 1 , ξ 2 )1 {|h(ξ1,ξ2)|>M} = +∞ 0 P h(ξ 1 , ξ 2 ) < -max(z, M ) dz - +∞ 0 P h(ξ 1 , ξ 2 ) > max(z, M ) dz = M P(h(ξ 1 , ξ 2 ) < -M ) -P(h(ξ 1 , ξ 2 ) > M ) + +∞ M P(h(ξ 1 , ξ 2 ) < -z) dz - +∞ M P(h(ξ 1 , ξ 2 ) > z) dz.
But, due to Item (iii) of Assumption 1, as x goes to infinity, we have

P(h(ξ 1 , ξ 2 ) > x) = c 0 x -β + o(x -β ), P(h(ξ 1 , ξ 2 ) < -x) = c 1 x -β + o(x -β ), +∞ x P(h(ξ 1 , ξ 2 ) > z) dz = c 0 x 1-β β -1 + o(x 1-β ), +∞ x P(h(ξ 1 , ξ 2 ) < -z) dz = c 1 x 1-β β -1 + o(x 1-β )
and ∀x > 0,

+∞ x P h(ξ 1 , ξ 2 ) > z + P h(ξ 1 , ξ 2 ) < -z dz ≤ ( L 0 ∞ + L 1 ∞ ) x 1-β β -1 .
Therefore, we obtain

(52) E h(ξ 1 , ξ 2 )1 {|h(ξ1,ξ2)|≤M} = M 1-β β β -1 (c 1 -c 0 ) + ǫ M ,
where lim M→+∞ ǫ M = 0 and sup M>0 ǫ M < ∞. • When β = 1, due to Item (vii) of Assumption 1, we have c 0 = c 1 and (52) holds also true.

• Assume now that β ∈ [1, 2). We will prove that lim δ→0 lim sup n→∞ E[(T n (δ)) 2 | F ] = 0. We have

E[(T n (δ)) 2 | F ] = x,y,x ′ ,y ′ ∈Z d 0 E[T n,x,y T n,x ′ ,y ′ | F ],
with T n,x,y := a -1 n h(ξ x , ξ y )ζ n,x,y 1 {|h(ξx,ξy)ζn,x,y|≤anδ} + a -β n (c 0c 1 )

βδ 1-β β -1 |ζ n,x,y | β - (recall that c 0 = c 1 when β = 1).
-Contribution of (x, y, x ′ , y ′ ) such that {x, y} ∩ {x ′ , y ′ } = ∅.

We set E 1 for the set of such (x, y, x ′ , y ′ ). Let (x, y, x ′ , y ′ ) ∈ E 1 . Since h(ξ x , ξ y ) and h(ξ x ′ , ξ y ′ ) are independent conditionally to F , we have

E[T n,x,y T n,x ′ ,y ′ | F ] = E[T n,x,y | F ]E[T n,x ′ ,y ′ | F ].
Now, due to (52), we have

x,y∈Z d 0 E[T n,x,y | F ] ≤ δ 1-β x,y∈Z d 0 a -β n |ζ n,x,y | β + ǫ anδ|ζn,x,y| -1 .
Now, due to (37), for every γ 0 > 0, if n is large enough, we have -Contribution of (x, y, x ′ , y ′ ) such that {x, y} = {x ′ , y ′ }.

Let us write E 2 for the set of such (x, y, x ′ , y ′ ). Observe that

(x,y,x ′ ,y ′ )∈E2 E[T n,x,y T n,x ′ ,y ′ | F ] ≤ 2 x,y∈Z d 0 E[T 2 n,x,y | F ].
First, using Item (iii) of Assumption 1, we notice that

a -2 n x,y∈Z d 0 E (h(ξ 1 , ξ 2 )ζ n,x,y ) 2 1 {|h(ξ1,ξ2)ζn,x,y|≤anδ} F = x,y∈Z d 0 δ 2 0 P √ z < a -1 n |h(ξ 1 , ξ 2 )ζ n,x,y | < δ F dz ≤ x,y∈Z d 0 δ 2 0 P √ z < a -1 n |h(ξ 1 , ξ 2 )ζ n,x,y | F dz ≤ ( L 0 ∞ + L 1 ∞ ) x,y∈Z d 0 δ 2 0 a -β n z -β 2 |ζ n,x,y | β dz ≤ ( L 0 ∞ + L 1 ∞ )a -β n x,y∈Z d 0 |ζ n,x,y | β δ 2(1-β 2 ) 1 -β 2 ≤ ( L 0 ∞ + L 1 ∞ ) G+ n δ 2-β 1 -β 2 . Therefore (54) lim δ→0 lim sup n→+∞ a -2 n x,y∈Z d 0 E (h(ξ 1 , ξ 2 )ζ n,x,y ) 2 1 {|h(ξ1,ξ2)ζn,x,y|≤anδ} F = 0.
Second, using (34) and the definition of Ñ * n and Rn , for every γ 0 > 0, for n large enough, we have

a -2β n x,y∈Z d 0 (c 0 -c 1 ) 2 β 2 δ 2-2β (β -1) 2 |ζ n,x,y | 2β - ≤ (c 0 -c 1 ) 2 β 2 δ 2-2β (β -1) 2 a -2β n R2 n ( Ñ * n ) 4β ≤ n -2 α 0 +2ǫ+4βǫ+γ0 δ 2-2β .
So, since ǫ > 0 satisfies (3 + 4β)ǫ < 1 α0 we have that

(c 0 -c 1 ) 2 β 2 δ 2-2β (β -1) 2 |ζ n,x,y | 2β - = 0.
Finally this shows

lim δ→0 lim sup n→+∞ (x,y,x ′ ,y ′ )∈E2 E[T n,x,y T n,x ′ ,y ′ | F ] = 0.
-Contribution of (x, y, x ′ , y ′ ) such that #({x, y} ∩ {x ′ , y ′ }) = 1.

Let us write E 3 for the set of such (x, y, x ′ , y ′ ). Observe that we have

(x,y,x ′ ,y ′ )∈E3 E[T n,x,y T n,x ′ ,y ′ | F ] = 4
x,y,z:x =y,x =z,y =z E T n,x,y T n,x,z | F * Assume that 1 ≤ β < 4/3. We set U n,x,y := a -1 n h(ξ x , ξ y )ζ n,x,y 1 {|h(ξx,ξy)ζn,x,y|≤anδ} . Observe that (56)

T n,x,y = U n,x,y + a -β n (c 0c 1 )

βδ 1-β β -1 |ζ n,x,y | β - (recall that we assume c 0 = c 1 if β = 1
) and that, due to (52), ( 57)

E[U n,x,y | F ] = a -β n δ 1-β |ζ n,x,y | β -(c 1 -c 0 ) β β -1 + ǫ anδ|ζn,x,y| -1 .
Now, (37) ensures that (58) lim n→+∞ sup

x,y ǫ anδ|ζn,x,y| -1 = 0.

Moreover, we observe that, due to (34) and to the definition of Ñ * n and of Rn , we have, for every γ 0 > 0 and every n large enough,

x,y,z∈Z d 0 a -2β n |ζ n,x,y | β |ζ n,x,z | β ≤ R3 n a -2β n Ñ * n 4β ≤ n -1 α 0 +3ǫ+4βε+γ0 . Now, since (3 + 4β)ǫ < 1 α0 we conclude that (59) lim sup n→+∞ x,y,z∈Z d 0 a -2β n |ζ n,x,z | β |ζ n,x,y | β = 0.
Observe moreover that, due to Item (iv) of Assumption 1, we have

E |U n,x,y U n,x,z || F ≤ (0,δ) 2 P(a -1 n |h(ξ 1 , ξ 2 )ζ n,x,y | > u, a -1 n |h(ξ 1 , ξ 3 )ζ n,x,z | > v| F ) dudv ≤ C 0 a -1 n |ζ n,x,y | + δ a -1 n |ζn,x,z| u -γ a -γ n |ζ n,x,y | γ du × a -1 n |ζ n,x,z | + δ a -1 n |ζn,x,z| v -γ a -γ n |ζ n,x,z | γ dv ≤ C 0 a -1 n |ζ n,x,y | 1 + δ 1-γ -a γ-1 n |ζ n,x,z | 1-γ 1 -γ a -γ n |ζ n,x,y | γ × a -1 n |ζ n,x,z | + δ 1-γ -a γ-1 n |ζ n,x,z | 1-γ 1 -γ a -γ n |ζ n,x,z | γ ≤ C δ a -2γ ′ n |ζ n,x,y ζ n,x,z | γ ′ where γ ′ = min(1, γ)
for n large enough and some C δ > 0. Indeed, due to (37) we have a -1 n sup x,y |ζ n,x,y | ≤ 1 for large n. Again using (37) and to the definition of Rn , for every γ 0 > 0, we have

x,y,z∈Z d 0 E |U n,x,y U n,x,z || F ≤ C δ R3 n a -2γ ′ n sup x,y |ζ n,x,y | 2γ ′ ≤ n 3 α 0 -4γ ′ α 0 β +7ε+γ0 ,
for n large enough. Recall that we have chosen ε such that

T n,x,y = a -1 n ζ n,x,y h (anδ|ζn,x,y| -1 ) (ξ x , ξ y ).
Due to this Item (vi), to the definition of Rn and to (37), for every γ 0 > 0, we have almost surely

x,y,z∈Z d 0

|E[T n,x,y T n,x,z | F ]| ≤ C ′ 0 a -2 n x,y,z∈Z d 0 |ζ n,x,y ζ n,x,z | a 2 n δ 2 |ζ n,x,y ζ n,x ′ ,y ′ | -1 -θ ′ ≤ δ -2θ ′ R3 n a -1 n ( Ñ * n ) 2 2(θ ′ +1) ≤ n 1 α 0 3-4(θ ′ +1) β +(4θ ′ +7)ε+γ0 , for n large enough. Since 1 α0 3 -4(θ ′ +1) β + (4θ ′ + 7)ε < 0, we obtain ∀δ > 0, lim sup n→+∞ (x,y,x ′ ,y ′ )∈E3 |E[T n,x,y T n,x,z | F ]| = 0.
So, finally, for β ∈ [1, 2), there exists C > 0 such that, for every nonnegative n and every δ > 0, we have lim sup

n→+∞ E[(T n (δ)) 2 ] ≤ Cδ 2-β .
Proof of Proposition 17. The following proof can be assembled from [START_REF] Feller | An introduction to probability theory and its applications[END_REF]. We will use the constants

I 0 := - ∞ 0
sin y y β dy and J 0 :=tan πβ 2 I 0 . Due to the exponential formula, we have

E e it {|x|≥δ} x dP(x) = exp {|x|≥δ} (e itx -1)(a1 {x>0} + b1 {x<0} )β|x| -β-1 dx = exp (a + b) +∞ δ cos(tx) -1 x β+1 β dx + i(a -b) +∞ δ sin(tx) x β+1 β dx
Assume first that β < 1. Due to [13, p. 568], we have Hence we have in that case that

lim δ→0 +∞ δ e itx -1 x β+1 β dx = -|t| β Γ(1 -β)e -iπβ 2 = |t| β (I 0 + iJ 0 ). So lim δ→0 E e it {|x|≥δ} x dP(x) = Φ a+b,a-b,β (t). Assume now that β = 1. Then lim δ→0 +∞ δ cos(tx) -1 x 2 dx = +∞ 0 cos(tx) -1 x 2 dx = |t| +∞ 0 cos(y) -1 y 2 dy = - π 2
lim δ→0 E exp it |x|>δ xdP(x) -(a -b) ∞ δ sin x x 2 dx = Φ a+b,a-b,1 (t).
Assume finally β > 1. Due to [13, p.568-569], we have

lim δ→0 ∞ δ e itx -1 -itx x β+1 β dx = +∞ 0 e itx -1 -itx x β+1 β dx = |t| β Γ(3 -β)e -iπβ 2 (2 -β)(β -1) = |t| β (I 0 + iJ 0 ). So lim δ→0 E e it {|x|≥δ} x dP(x)-it(a-b)β δ 1-β β-1
= Φ a+b,a-b,β (t).

Tightness

Here we treat case α 0 > 1 (i.e. the case where (S n ) n is recurrent and α > d 0 = 1). The tightness proof follows essentially the one given in Kesten and Spitzer [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF]. We need the following lemma from [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF].

Lemma 19 (Lemma 1 of [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF]). For all ǫ > 0 there exists some A > 0 such that for all t ≥ 1 one has

P ∃x ∈ Z : |x| > At 1/α and N t (x) > 0 ≤ ǫ.
Lemma 20. We have

E x∈Z N 2 n (x) = O(n 2-1 α ) and E x∈Z N 2 n (x) 2 = O(n 4-2 α ). ( 61
)
Proof. The first one is formula (2.13) from [START_REF] Kesten | A limit theorem related to an new class of self similar processes[END_REF] and the second one can be found in [

P U n t1 -U n t2 > η = 0.
Fix some ǫ > 0. Due to Lemma 19, we fix A > 0 large enough such that

P ∃x ∈ Z with |x| > An 1/α and N ⌊nT ⌋ (x) > 0 ≤ ǫ 4 . (62)
Choose some ρ > 0 such that for all n ∈ N one has .

The inequality (63) now becomes

(65) 9A 2 n 2/α P h(ξ 1 , ξ 2 ) = h(ξ 1 , ξ 2 ) ≤ ǫ 4 .
Lemma 22. There exists a constant C = C(ρ, β) > 0 such that for all n ≥ 1 one has

E h(ξ 1 , ξ 2 ) ≤ Cn (1-β) 2 αβ . (66) Proof. For β < 1, we have E h(ξ 1 , ξ 2 ) ≤ ρn 2 αβ 0 P (|h(ξ 1 , ξ 2 )| > x) dx ≤ C ρn 2 αβ 1 x -β dx + 1 = Cx 1-β ρn 2/αβ 1 + 1 ∼ Cn 2 αβ (1-β)
where C > 0 is some suitable constant. For β ∈ (1, 2), this comes from (52). For β = 1, as noticed previously, this comes from Item (vii) of Assumption 1.

Now we define

E n := n -2δ E   x,y∈Z N n (x)N n (y) h(ξ x , ξ y )   .
Since the scenery and the random walk are independent, we compute

E n = n -2δ E   x,y∈Z N n (x)N n (y)E h(ξ x , ξ y )   = n -2δ n 2 E h(ξ 1 , ξ 2 ) ≤ Cn -2+ 2 α -2 αβ n 2 n (1-β) 2 αβ = C,
due to Lemma 22. Thus the sequence E n stays bounded as n → ∞. Further, let

Ū n t := n -2δ
x,y∈Z

N ⌊nt⌋ (x)N ⌊nt⌋ (y) h(ξ x , ξ y ) -E h(ξ x , ξ y ) .
It then follows

U n t -Ū n t -t 2 E n = n -2δ
x,y∈Z

N ⌊nt⌋ (x)N ⌊nt⌋ (y) h(ξ x , ξ y ) -h(ξ x , ξ y ) + n -2δ ⌊nt⌋ 2 E h(ξ 1 , ξ 2 ) -n 2 t 2 E h(ξ 1 , ξ 2 ) . Since we have that E[ h(ξ 1 , ξ 2 )] = O(n (1-β) 2 αβ ) and ⌊nt⌋ 2 -n 2 t 2 = O(n) the second term is of the order n -2δ O(n (1-β) 2 αβ )(⌊nt⌋ 2 -n 2 t 2 ) = n -2 O(n) = O(n -1 ).
This implies with inequalities (62) and (65) that

lim sup n→∞ P sup 0≤t≤T U n t -Ū n t -t 2 E n > η 2 ≤ lim sup n→∞ P   n -2δ
x,y∈Z

N ⌊nT ⌋ (x)N ⌊nT ⌋ (y) h(ξ x , ξ y ) -h(ξ x , ξ y ) > η 4   ≤ lim sup n→∞ P   x,y∈Z N ⌊nT ⌋ (x)N ⌊nT ⌋ (y) h(ξ x , ξ y ) -h(ξ x , ξ y ) = 0   ≤ lim sup n→∞ P ∃x, y ∈ Z : |x|, |y| ≤ An 1/α , h(ξ x , ξ y ) = h(ξ x , ξ y ) + lim sup n→∞ P ∃x ∈ Z : |x| > An 1/α , N ⌊nT ⌋ (x) > 0 ≤ lim sup n→∞ (3An 1/α ) 2 P h(ξ 1 , ξ 2 ) = h(ξ 1 , ξ 2 ) + ǫ 4 ≤ ǫ 2 .
It is now sufficient to prove that

lim sup n→∞ lim κ↓0 sup 0≤t1,t2≤T :|t1-t2|≤κ P Ū n t1 -Ū n t2 > η 2 = 0.
For this we prove for all T ≥ t > s ≥ 0 that

E Ū n t -Ū n s 2 ≤ C(t -s) 2-2 α . ( 67
)
If we use the notation h0 (ξ x , ξ y ) := h(ξ x , ξ y ) -E h(ξ x , ξ y ) then we have

E Ū n t -Ū n s 2 = n -4δ E x,y N ⌊nt⌋ (x) N ⌊nt⌋ (y) -N ⌊ns⌋ (y) h0 (ξ x , ξ y ) + x,y N ⌊nt⌋ (x) -N ⌊ns⌋ (x) N ⌊ns⌋ (y) h0 (ξ x , ξ y ) 2 ≤ 2n -4δ E x,y N ⌊nt⌋ (x) N ⌊nt⌋ (y) -N ⌊ns⌋ (y) h0 (ξ x , ξ y ) 2 + 2n -4δ E x,y N ⌊nt⌋ (x) -N ⌊ns⌋ (x) N ⌊ns⌋ (y) h0 (ξ x , ξ y ) 2 .
We continue the computation with the first of the two terms. In the following we condition with respect to G = σ(S n ; n ∈ N). We make use of the assumption h(x, x) = 0 and the fact that if x, y, u, v are all distinct then h0 (ξ x , ξ y ) and h0 (ξ u , ξ v ) are independent and centered and we write

E x,y N ⌊nt⌋ (x) N ⌊nt⌋ (y) -N ⌊ns⌋ (y) h0 (ξ x , ξ y ) 2 G ≤ A + B + C + D with A := x,y N 2 ⌊nt⌋ (x) N ⌊nt⌋ (y) -N ⌊ns⌋ (y) 2 E h2 0 (ξ 1 , ξ 2 ) G , B := x,y,z N ⌊nt⌋ (x)N ⌊nt⌋ (z) N ⌊nt⌋ (y) -N ⌊ns⌋ (y)
≤ C ′ n 2 n 2-1 α (t -s) 2-1 α n -3 α + 4 αβ = (t -s) 2-1 α O(n 4δ
). Again we see ≤ C ′ (nt) 1-1 2α (n(ts)) All those inequalities together prove that there exists some constant K > 0 such that for (ts) < κ < 1 one has

E [C] = (⌊nt⌋ -⌊ns⌋) 2 E x N 2 ⌊nt⌋ (x) Cov h(ξ 1 , ξ 2 ), h(ξ 2 , ξ 3 ) ≤ n 2 (t -s) 2 n 2-1 α T 2-1 α n -3 α + 4 αβ = (t -s) 2 O(n 4δ ).
E Ū n t -Ū n s 2 ≤ K(t -s) 2-2 α .
This finishes the tightness proof.

Lemma 23. There is some constant C > 0 such that Cov h(ξ 1 , ξ 2 ), h(ξ 1 , ξ 3 ) ≤ C ′ n -3 α + 4 αβ .

Proof. We first do the case β < 4 3 . Note that by Assumption 1 part (iv) for some γ > 3β 4 (γ = 1), we have We extend this definition by linearity to any linear combination H of such indicator functions. Observe that, if H = µ j=1 h j 1 Aj where (A j ) j is a family of pairwise disjoint rectangles and where h j ∈ R, then the characteristic function of R 2 H(x, y) dZ x,y is given by ∀z ∈ R, E exp iz -dxdy,β (z)). Proposition 25. (see [START_REF] Khoshnevisan | Level sets of the stochastic wave equation driven by a symmetric Lévy noise[END_REF]) Let H be a continuous compactly supported function from R 2 to R. Let (H n ) n be a sequence of linear combination of indicators over rectangles converging pointwise to H. Assume moreover that (H n ) n is a family of uniformly bounded functions with support in a same compact. Then the sequence R 2 H n (x, y) dZ(x, y) n converges in probability to a random variable with characteristic function Φ (c0+c1) R 2 |H(x,y)| β + dxdy,(c0-c1) R 2 |H(x,y)| β -dxdy,β . For a continuous compactly supported H : R 2 → R, we define R 2 H(x, y) dZ(x, y) as the limit in probability given by Proposition 25 (observe that the limit does not depend on the choice of (H n ) n ).

E | h(ξ 1 , ξ 2 ) h(ξ 1 , ξ 3 )| =
Proof of Proposition 25. To prove the convergence in probability, it is enough to prove that Observe that, for every real number z, we have

E exp iz

  #{x ∈ Z : Ñn,tm (x) > 0} = Rn and (31) G± n = n -2βδ x,y∈Z m i=1 θ i Ñn,ti (x) Ñn,ti (y) β ± .

  Ñn,ti (x) Ñn,ti (y) β ± which has the same distribution as

  ζn,x,y(ω)h(ξx,ξy) | F ] = Φ (c0+c1) G+ (ω),(c0-c1) G-(ω),β (1), for P-almost every ω in Ω and lim n→+∞ E e ia -1 n x,y ζn,x,yh(ξx,ξy) = E Φ (c0+c1) G+ ,(c0-c1) G-,β (1) .

2 α

 2 0 β +2ε+γ0 . Combining this with lim n→+∞ G+ n = G+ and with lim M→+∞ ǫ M = 0, we obtain (53) lim sup n→+∞ x,y∈Z d 0 E[T n,x,y | F ] = 0, since βǫ < 1/α 0 . This implies ∀δ > 0, lim sup n→+∞ (x,y,x ′ ,y ′ )∈E1 E[T n,x,y T n,x ′ ,y ′ | F ] = 0.

9A 2 n

 2 2/α P |h(ξ 1 , ξ 2 )| > ρn since we have, by Item (iii) of Assumption 1, that lim u→∞ u β P(h(ξ 1 , ξ 2 ) ≥ u) = c 0 and lim u→∞ u β P(h(ξ 1 , ξ 2 ) ≤ -u) = c 1 . (64) Define h(x, y) := h(x, y)1 {|h(x,y)|≤ρn 2 αβ }

2 E

 2 | h0 (ξ 1 , ξ 2 ) h0 (ξ 2 , ξ 3 )| G , C := x,y,z N 2 ⌊nt⌋ (x) N ⌊nt⌋ (y) -N ⌊ns⌋ (y) N ⌊nt⌋ (z) -N ⌊ns⌋ (z) E | h0 (ξ 1 , ξ 2 ) h0 (ξ 2 , ξ 3 )| G .andD := 2 x,x ′ ,y N ⌊nt⌋ (x ′ )N ⌊nt⌋ (x) N ⌊nt⌋ (y) -N ⌊ns⌋ (y) N ⌊nt⌋ (x) -N ⌊ns⌋ (x) E | h0 (ξ 1 , ξ 2 ) h0 (ξ 2 , ξ 3 )| G .The Markov property together with Lemma 20 and Lemma 23 below implyE[B] ≤ T 2 n 2 E x N 2 ⌊nt⌋-⌊ns⌋ (x) Cov h(ξ 1 , ξ 2 ), h(ξ 2 , ξ 3 )

Further, we haveN

  by Cauchy-Schwarz thatE x N ⌊nt⌋ (x) N ⌊nt⌋ (x) -N ⌊ns⌋ (x) ⌊nt⌋ (x) -N ⌊ns⌋ (x)

PPC 0 2 = 3 α + 4 αβ ). Now assume β ≥ 4 3 . 2 ≤+ O n - 4 αβ (β- 1 ) 1 . 2 α + 4 αβ.Proof. We have E h(ξ 1 , ξ 2 )) 2 = ρn 2 αβ 0 P(| h(ξ 1 , ξ 2 )| 2 ≥ s) ds = ρn 2 αβ 0 P= O(n 2 αβ ( 2 2 1A

 023432411241201220222 | h(ξ 1 , ξ 2 )| > s, | h(ξ 1 , ξ 3 )| > t dsdt = (|h(ξ 1 , ξ 2 )| > s, |h(ξ 1 , ξ 3 )| > t) dsdt ≤ (max(1, s) max(1, t)) -γ dsdt O(n -3 α + 4 αβ )Due to Lemma 22 this impliesCov h(ξ 1 , ξ 2 ), h(ξ 1 , ξ 3 ) = O(n -By (52) and Item (vi) of Assumption 1, we have for M n := ρn2 αβ that Cov h(ξ 1 , ξ 2 ), h(ξ 1 , ξ 3 ) = |Cov (h Mn (ξ 1 , ξ 2 ), h Mn (ξ 1 , ξ 3 ))| ≤ |E [h Mn (ξ 1 , ξ 2 )h Mn (ξ 1 , ξ 3 )]| + |E[h Mn (ξ 1 , ξ 2 )]| O n -4θ ′ αβ Lemma 24. We have E h(ξ 1 , ξ 2 )) 2 = O n -(| h(ξ 1 , ξ 2 )| ≥ u)2u du -β) ), since 2uP(| h(ξ 1 , ξ 2 )| ≥ u) ∼ 2(c 0 + c 1 )u 1-β as u goes to infinity. • Z 0,0 = 0; • for any family (A k = [a k , b k ]×[a ′ k , b ′ k ]) k of pairwise disjoint rectangles (with a k < b k and a ′ k < b ′ k ), the family of increments (Z b k ,b ′ k + Z a k ,a ′ k -Z a k ,b ′ k -Z b k ,a ′ k ) k is a family of independent random variables; • for any rectangle A = [a, b] × [a ′ , b ′ ] (with a < b and a ′ < b ′ ), the characteristic function of the increment Z b,b ′ + Z a,a ′ -Z a,b ′ -Z b,a ′ is Φ (c0+c1)λ(A),(c0-c1)λ(A),β, where λ is the Lebesgue measure on R 2 and where we used the notation introduced in (4). For any rectangle A = [a, b] × [a ′ , b ′ ] (with a < b and a ′ < b ′ ), we define the stochastic integral of 1 A with respect to the Lévy process as the increment of Z in this rectangle, i.e. (68) R dZ x,y := Z b,b ′ + Z a,a ′ -Z a,b ′ -Z b,a ′ .

R 2 HE exp izh j R 2 1 j=1 Φ

 22j=1 (x, y) dZ x,y = µ j=1 Aj (x, y) dZ x,y = µ j=1 Φ (c0+c1)λ(Aj ),(c0-c1)λ(Aj ),β (zh j ) = µ (c0+c1)|hj| β + λ(Aj ),(c0-c1)|hj| β -λ(Aj ),β (z) = Φ (c0+c1) µ j=1 |hj| β + λ(Aj ),(c0-c1) µ j=1 |hj| β -λ(Aj ),β(z) and so by (69) ∀z ∈ R, E exp iz

R 2 H

 2 (x, y) dZ x,y = Φ (c0+c1) R 2 |H(x,y)| β + dxdy,(c0-c1) R 2 |H(x,y)| β

  (x, y) -H m (x, y)) dZ x,y = 1.

R 2 ( 1 =

 21 H n (x, y) -H m (x, y)) dZ x,y -Φ (c0+c1) R 2 |Hn(x,y)-Hm(x,y)| β + dxdy,(c0-c1) R 2 |Hn(x,y)-Hm(x,y)| β + dxdy,β (z) -1 ≤ C R 2 |H n (x, y) -H m (x, y)| β dxdy(|c 0 + c 1 | + |c 0c 1 |)|z| β , using the fact that |e -a+ibe -a ′ +ib ′ | ≤ |aa ′ | + |bb ′ |for any real numbers a, b, a ′ , b ′ such that a > 0 and a ′ > 0. Since (H n ) n converges pointwise and is uniformly bounded, we obtain (70) by the Lebesgue dominated convergence theorem (recall that (H n ) n is a sequence of uniformly bounded functions supported in a same compact). Now the characteristic function of the limit in probability

R 2 H

 2 (x, y) dZ(x, y) is given byE exp iz

R 2 H

 2 (x, y) dZ(x, y) = lim n→+∞ E exp iz

R 2 H

 2 n (x, y) dZ(x, y)= lim n→+∞ Φ (c0+c1) R 2 |Hn(x,y)| β + dxdy,(c0-c1) R 2 |Hn(x,y)| β -dxdy,β (z)) = Φ (c0+c1) R 2 |H(x,y)| β + dxdy,(c0-c1) R 2 |H(x,y)| β -dxdy,β (z)),for every real number z.

  T n,x ′ ,y ′ | F ] = 0. * Assume now that β ≥ 4 3 . Observe that, with the notation of Item (vi) of Assumption 1, we have

	we obtain			3 α0 -4γ ′ α0β + 7ε < 0. Hence,
	(60)	∀δ > 0, lim sup n→+∞ x,y,z	E[|U n,x,y U n,x,z |] = 0.
	Now putting (56), (57), (58), (59) and (60) all together, we conclude that
	∀δ > 0, lim sup n→+∞ (x,y,x ′ ,y ′ )∈E3	E[T n,x,y

  15, Lemma 2.1].

	Proposition 21. The sequence of stochastic processes
	U n t := n -2δ
			x,y∈Z
	Proof. It is sufficient to show that
	lim sup n→∞	lim κ↓0	sup 0≤t1,t2≤T :|t1-t2|≤κ

N ⌊nt⌋ (x)N ⌊nt⌋ (y)h(ξ x , ξ y ); t ≥ 0 is tight in C(0, T ) with sup-norm.

  1-1 2α . ⌊nt⌋ (x) N ⌊nt⌋ (x) -N ⌊ns⌋ (x) Cov h(ξ 1 , ξ 2 ), h(ξ 2 , ξ 3 ) ≤ C ′′ n(n(ts))(nt) 1-1 2α (n(ts)) 1-1 2α n -3 α + 4 αβ For ts < κ < 1 this is smaller than C ′′ (ts) 2-2 α O(n 4δ). Finally for A, due to Lemma 24 below, we have

	Now Lemma 23 implies
	E [D] ≤ [nt]([nt) -[ns])E
					2		
	E [A] ≤	E		x	N 2 nt (x)	 E		y	N 2 n(t-s) (y)

x

N 2   Var h(ξ 1 , ξ 2 ) ≤ C ′′ O((tn) 4-2 α )O(((ts)n) 4-2 α )E[( h(ξ 1 , ξ 2 )) 2 ] ≤ C ′′′ (ts) 2-1 α n 4-2 α n -2 α + 4 αβ ≤ C ′′′ (ts) 2-1 α n 4δ .

ANR project MEMEMO2 (ANR-10-BLAN-0125). 1 to simplify notations, for every k ∈ Z, we write ξ k for ξ (k,...,

Appendix A. Stochastic integral with respect to the Lévy sheet Z In this section, following [START_REF] Khoshnevisan | Level sets of the stochastic wave equation driven by a symmetric Lévy noise[END_REF], we give a simple construction of stochastic integral with respect to the β-stable Lévy sheet Z. In [START_REF] Khoshnevisan | Level sets of the stochastic wave equation driven by a symmetric Lévy noise[END_REF], Khoshnevisan and Nualart considered general Lévy sheet with symmetric distributions. Therefore their results apply to the β-stable Lévy sheet Z only if c 0 = c 1 . Nevertheless, we will see that their construction is expansible when c 0 = c 1 .

Let us recall that Z satisfies the following properties:

[23] F. Spitzer, Principles of Random Walks. Van Nostrand, Princeton, NJ (1964).

Université de Brest, UMR CNRS 6205, Laboratoire de Mathématique de Bretagne Atlantique, 6 avenue Le Gorgeu, 29238 Brest cedex, France E-mail address: Brice.Franke@univ-brest.fr