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

Nadine Guillotin-Plantard

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

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

Françoise Pène

On the local time of random processes in random scenery

Keywords: 2000 Mathematics Subject Classification. 60F05, 60F17, 60G15, 60G18, 60K37 Random walk in random scenery, local limit theorem, local time, level

Random walks in random scenery are processes defined by Zn := n k=1 ξX 1 +...+X k , where basically (X k , k ≥ 1) and (ξy, y ∈ Z) are two independent sequences of i.i.d. random variables. We assume here that X1 is Z-valued, centered and with finite moments of all orders. We also assume that ξ0 is Z-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that (n -3/4 Z [nt] , t ≥ 0) converges in distribution as n → ∞ toward some self-similar process (∆t, t ≥ 0) called Brownian motion in random scenery. In a previous paper, we established that P(Zn = 0) behaves asymptotically like a constant times n -3/4 , as n → ∞. We extend here this local limit theorem: we give a precise asymptotic result for the probability for Z to return to zero simultaneously at several times. As a byproduct of our computations, we show that ∆ admits a bi-continuous version of its local time process which is locally Hölder continuous of order 1/4 -δ and 1/6 -δ, respectively in the time and space variables, for any δ > 0. In particular, this gives a new proof of the fact, previously obtained by Khoshnevisan, that the level sets of ∆ have Hausdorff dimension a.s. equal to 1/4. We also get the convergence of every moment of the normalized local time of Z toward its continuous counterpart.

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

We suppose that ξ 0 is centered, with finite second moment σ 2 := E[ξ 2 0 ]. The sequence ξ is called the random scenery.

The random walk in random scenery Z is then defined for all n ≥ 1 by

Z n := n-1 k=0 ξ S k .
For motivation in studying this process and in particular for a description of its connections with many other models, we refer to [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF][START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF][START_REF] Doussal | Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields[END_REF] and references therein. 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, and important, to see that Z n can be rewritten as Z n = y ξ y N n (y).

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] proved the following functional limit theorem :

n -3/4 Z nt , t ≥ 0 (L) =⇒ n→∞ (σ∆ t , t ≥ 0) , (1)
where

• Z s := Z n + (s -n)(Z n+1 -Z n ), for all n ≤ s ≤ n + 1,
• ∆ is defined by

∆ t := +∞ -∞ L t (x) dβ x ,
with (β x ) x∈R a standard Brownian motion and (L t (x), t ≥ 0, x ∈ R) a jointly continuous in t and x version of the local time process of some other standard Brownian motion (B t ) t≥0 independent of β.

The process ∆ is known to be a continuous (3/4)-self-similar process with stationary increments, and is called Brownian motion in random scenery. It can be seen as a mixture of stable processes, but it is not a stable process.

Let now ϕ ξ denote the characteristic function of ξ 0 and let d be such that {u : |ϕ ξ (u)| = 1} = (2π/d)Z. In [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF] we established the following local limit theorem :

P Z n = n 3 4 x = dσ -1 p 1,1 (x/σ) n -3 4 + o(n -3 4 ) if P nξ 0 -n 3 4 x ∈ dZ = 1 0 otherwise, (2)
with

p 1,1 (x) := 1 √ 2π E ||L 1 || -1 2 e -||L 1 || 2 2 x 2 /2 ,
and ||L 1 || 2 := R L 2 1 (y) dy 1/2 the L 2 -norm of L 1 . In the particular case when x = 0, we get

P (Z n = 0) = dσ -1 p 1,1 (0) n -3 4 + o(n -3 4 ) if n ∈ d 0 Z 0 otherwise, (3)
with d 0 := min{m ≥ 1 : ϕ ξ (2π/d) m = 1}. 1 Actually the results mentioned above were proved in the more general case when the distributions of the ξ y 's and X k 's are only supposed to be in the basin of attraction of stable laws (see [START_REF] Castell | Limit theorems for one and two-dimensional random walks in random scenery[END_REF], [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF] and [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] for details).

1.2. Statement of the results. T 1 ,...,T k .

Our first result is the following 1 Recall that, for every n ≥ 0, we have P(nξ0 ∈ dZ) > 0 ⇐⇒ P(nξ0 ∈ dZ) = 1 ⇐⇒ n ∈ d0Z.

Theorem 1. For any k ≥ 1, there exist constants c > 0 and C > 0, such that

c k i=1 (T i -T i-1 ) -3/4 ≤ C T 1 ,...,T k ≤ C k i=1 (T i -T i-1 ) -3/4 ,
for every 0 < T 1 < • • • < T k , with the convention that T 0 = 0.

The most difficult (and interesting) part here is the upper bound. The lower bound is obtained directly by using the scaling property of the local time of Brownian motion and the well-known Gram-Hadamard inequality. Concerning the upper bound, we will give more details about its proof in a moment, but let us stress already that even for k = 2 the result is not immediate (whereas when k = 1 it follows relatively easily from the Cauchy-Schwarz inequality and some basic properties of the Brownian motion, see for instance [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF]).

A first corollary of this result is the following: Corollary 2. For all k ≥ 1 and all 0 < T 1 < • • • < T k , the random variable (∆ T 1 , . . . , ∆ T k ) admits a continuous density function, denoted by p k,T 1 ,...,T k , which is given by

p k,T 1 ,...,T k (x) := (2π) -k 2 E D -1/2
T 1 ,...,T k exp -

1 2 M -1 T 1 ,...,T k x, x for all x ∈ R k .
Theorem 1 also shows that, for every t ≥ 0, k ≥ 1 and x ∈ R,

M k,t (x) := [0,t] k p k,T 1 ,...,T k (x, . . . , x) dT 1 . . . dT k , (4)
is finite. Define now the level sets of ∆ as the sets of the form

∆ -1 (x) := {t ≥ 0 : ∆ t = x},
for x ∈ R. We can then state our main application of Theorem 1, which can be deduced by standard techniques:

Theorem 3. There exists a nonnegative process (L t (x), x ∈ R, t ≥ 0), such that (i) a.s. the map (t, x) → L t (x) is continuous and nondecreasing in t. Moreover for any δ > 0, it is locally Hölder continuous of order 1/4δ, in the first variable, and of order 1/6δ, in the second variable, (ii) a.s. for any measurable ϕ : R → R + , and any t ≥ 0, t 0 ϕ(∆ s ) ds = R ϕ(x) L t (x) dx, (iii) for any T > 0, we have the scaling property:

(L t T (x), t ≥ 0, x ∈ R) (d) = (T 1/4 L t (x T -3/4 ), t ≥ 0, x ∈ R).
(iv) for any x ∈ R, k ≥ 1, and t > 0, the k-th moment of L t (x) is finite and

E L t (x) k = M k,t (x), (5)
(v) a.s. for any x ∈ R, the support of the measure

d t L t (x) is contained in ∆ -1 (x).
The random variable L t (x) is called the local time of ∆ in x at time t.

We believe that the exponents 1/4 and 1/6 in Part (i) are sharp. One reason is that our proof gives the right critical exponents in the case of the Brownian motion. Another heuristic reason comes from a result proved by Dombry & Guillotin [START_REF] Dombry | Discrete approximation of a stable self-similar stationary increments process[END_REF], saying that the sum of n i.i.d copies of the process ∆ converges under appropriate normalization, towards a fractional Brownian motion with index 3/4. But the Hölder continuity critical exponents of the local time of the latter process are exactly equal to 1/4 in the time variable, and 1/6 in the space variable.

Let us point out that an original feature of this theorem is that it gives strong regularity properties of the local time of a process which is neither Markovian nor Gaussian, whereas usually similar results are obtained when at least one of these conditions is satisfied (see for instance [START_REF] Geman | Occupation densities[END_REF][START_REF] Marcus | Markov processes, Gaussian processes, and local times[END_REF]).

We should notice now that previously only the existence of a process satisfying (ii), [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF] for k ≤ 2 and (v) was known, see [START_REF] Xiao | The Hausdorff dimension of the level sets of stable processes in random scenery[END_REF]. The original motivation of [START_REF] Xiao | The Hausdorff dimension of the level sets of stable processes in random scenery[END_REF] was in fact to study the Hausdorff dimension of the level sets of ∆. Khoshnevisan and Lewis conjectured in [START_REF] Khoshnevisan | Iterated Brownian motion and its intrinsic skeletal structure[END_REF] that their Hausdorff dimension was a.s. equal to 1/4, for every x ∈ R. In [START_REF] Xiao | The Hausdorff dimension of the level sets of stable processes in random scenery[END_REF] Xiao proved the result for almost every x, and left open the question to know whether this was true for every x. This has been later proved by Khoshnevisan in [START_REF] Khoshnevisan | The codimension of the zeros of a stable process in random scenery[END_REF]. With Theorem 3 we can now give an alternative proof, which follows the same lines as standard ones in the case of the Brownian motion :

Corollary 4 (Khoshnevisan [START_REF] Khoshnevisan | The codimension of the zeros of a stable process in random scenery[END_REF]). For every x ∈ R, the Hausdorff dimension of ∆ -1 (x) is a.s. equal to 1/4.

Actually Xiao and Khoshnevisan proved their result in the more general setting where the Brownian motion B is replaced by a stable process of index α ∈ [START_REF] Billingsley | Probability and measure[END_REF][START_REF] Borodin | A limit theorem for sums of independent random variables defined on a recurrent random walk. (Russian)[END_REF]. But at the moment it does not seem straightforward for us to adapt our proof to this case. Now let us give some rough ideas of the proof of Theorem 1. The first thing we use is that M T 1 ,...,T k is a Gram matrix, and so there is nice formula for its smallest eigenvalue [START_REF] Dombry | Discrete approximation of a stable self-similar stationary increments process[END_REF], which shows that to get a lower bound, it suffices to prove that the term L T 1 /T 3/4 1 is far in L 2 -norm from the vector space generated by the terms (L T j -L T j-1 )/(T j -T j-1 ) 3/4 , for j ≥ 2. Now by scaling we can always assume that T 1 = 1. Next by using the Hölder regularity of the process L, we can replace the L 2 -norm by the L ∞ -norm, which is much easier to control. Then we use the Ray-Knight theorem, which says that, if instead of considering the term L 1 we consider L τ , with τ some appropriate random time, then we get a Markov process. It is then possible to prove that with high probability, this process is far in L ∞ -norm from any finite dimensional affine space, from which the desired result follows.

1.2.2. Random walk in random scenery. Our first result is a multidimensional extension of our previous local limit theorem. We state it only for return probabilities to 0, to simplify notation, but it works exactly the same is we replace 0 by [n 3/4 x], for some fixed x = 0. Theorem 5. Let k ≥ 1 be some integer and let 0 < T 1 < • • • < T k , be k fixed positive reals. Then for any n ≥ 1,

• If [nT i ] ∈ d 0 Z, for all i ≤ k, then P Z [nT 1 ] = • • • = Z [nT k ] = 0 = dσ -1 k p k,T 1 ,...,T k (0, . . . , 0) n -3k/4 + o(n -3k/4 ). • Otherwise P Z [nT 1 ] = • • • = Z [nT k ] = 0 = 0.
Moreover, for every k ≥ 1 and every θ ∈ (0, 1), there exists C = C(k, θ) > 0, such that

P [Z n 1 = • • • = Z n 1 +•••+n k = 0] ≤ C (n 1 . . . n k ) -3/4 , for all n ≥ 1 and all n 1 , . . . , n k ∈ [n θ , n].
As an application we can prove that the moments of the local time of Z converge toward their continuous counterpart. More precisely, for z ∈ Z, define the local time of Z in z at time n by:

N n (z) := #{m = 1, . . . , n : Z m = z}.
Then Theorem 5 together with the Lebesgue dominated convergence theorem give Corollary 6. For all k ≥ 1,

E N n (0) k ∼ d σd 0 k M k,1 (0) n k/4 ,
as n → ∞, with M k,1 (0) as in [START_REF] Castell | Limit theorems for one and two-dimensional random walks in random scenery[END_REF].

A natural question now is to know if we could not deduce from this corollary the convergence in distribution of the normalized local time N n (0)/n 1/4 toward L 1 (0). To this end, we should need to know that the law of L 1 (0) is determined by the sequence of its moments. Since this random variable is nonnegative, a standard criterion ensuring this, called Carleman's criterion, is the condition:

k M k,1 (0) -1 2k = ∞.
In particular a bound for M k,1 (0) in k 2k would be sufficient. However with our proof, we only get a bound in k ck , for some constant c > 0. We can even obtain some explicit value for c, but unfortunately it is larger than 2, so this is not enough to get the convergence in distribution.

Note that this question is directly related to the question of the dependence in k of the constant C in Theorem 1, which we believe is an interesting question for other problems as well, such as the problem of large deviations for the process L (see for instance [START_REF] Chen | Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes[END_REF] in which the case of the fractional Brownian motion is considered).

Another interesting feature of Theorem 5 is that it gives an effective measure of the asymptotic correlations of the increments of Z. Indeed, if we assume to simplify that k = 2, σ = 1 and d = 1, then (2) and Theorem 5 (actually its proof) show that

P(Z n+m -Z n = 0 | Z n = 0) P(Z n+m -Z n = 0) -→ E ||L 1 || 2 2 || L t || 2 2 -L 1 , L t 2 -1/2 E ||L 1 || 2 || L t || 2 -1 , (6)
as n → ∞ and m/n → t, for some t > 0, where L and L are the local time processes of two independent standard Brownian motions. In particular the limiting value in ( 6) is larger than one, which means that the process is asymptotically more likely to come back to 0 at time n + m, if we already know that it is equal to 0 at time n.

The general scheme of the proof of Theorem 5 is quite close from the one used for the proof of (3) in [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF]. However, in addition to Theorem 1 which is needed here and which is certainly the main new difficulty, some other serious technical problems appear in the multidimensional setting. In particular at some point we use a result of Borodin [START_REF] Borodin | On the character of convergence to Brownian local time. II[END_REF] giving a strong approximation of the local time of Brownian motion by the random walk local time. This also explains why we need stronger hypothesis on the random walk here. Now concerning the scenery, it is not clear if we can relax the hypothesis of finite second moment, since we strongly use that the characteristic function of (∆ T 1 , ..., ∆ T k ), takes the form

ψ(θ 1 , ..., θ k ) = E e -k i,j=1 a i,j θ i θ j ,
with (a i,j ) i,j some (random) positive symmetric matrix.

Finally let us mention that in the proof of Theorem 5, we use the following result, which might be interesting on its own. It is a natural multidimensional extension of a result of Kesten and Spitzer [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] on the convergence in distribution of the normalized self-intersection local time of the random walk. Proposition 7. Let k ≥ 1 be given and let

T 1 < • • • < T k , be k positive reals. Then n -3/2 N n 1 +•••+n i , N n 1 +•••+n j 1≤i,j≤k (L) =⇒ L T i , L T j 1≤i,j≤k ,
as n → ∞, and (n 1 + • • • + n i )/n → T i , for all i ≥ 1, and where for all p, q, N p , N q := y∈Z N p (y)N q (y).

The paper is organized as follows. In Section 2, we give a short proof of Corollary 2. In Section 3 we prove Theorem 1. Then in Section 4, we explain how one can deduce Theorem 3 and Corollary 4 from it. Section 5 is devoted to the proof of Theorem 5 and Section 6 to the proof of Corollary 6. Finally, in Section 7, we give a proof of Proposition 7.

We also mention some notational convention that we shall use: if X is some random variable and A some set, then E[X, A] will mean E[X1 A ].

Proof of corollary 2

Let k ≥ 1 be given and let T 1 < • • • < T k , be some positive reals. The characteristic function ψ T 1 ,...,T k of (∆ T 1 , . . . , ∆ T k ) (with the convention T 0 = 0) is given by

ψ T 1 ,...,T k (θ) = E   exp   - 1 2 R k i=1 θ i L T i (u) 2 du     = E exp - 1 2 M T 1 ,...,T k θ, θ ,
with θ := (θ 1 , . . . , θ k ). In particular this function is non-negative. Moreover, a change of variables (this change is possible since D T 1 ,...,T k is almost surely non null, thanks to Theorem 1) gives

R k ψ T 1 ,...,T k (θ) dθ = C T 1 ,...,T k R k e -1 2 k i=1 u 2 i du 1 . . . du k = (2π) k 2 C T 1 ,...,T k < ∞.
This implies, see the remark following Theorem 26.2 p.347 in [START_REF] Billingsley | Probability and measure[END_REF], that (∆ T 1 , . . . , ∆ T k ) admits a continuous density function p k,T 1 ,...,T k , given by

p k,T 1 ,...,T k (x) = 1 (2π) k R k e -i θ,x ψ T 1 ,...,T k (θ) dθ = (2π) -k 2 E D -1 2 T 1 ,...,T k exp - 1 2 M -1 T 1 ,...,T k x, x ,
which was the desired result.

Proof of Theorem 1

Let k ≥ 1 and 0 < T 1 < ... < T k , be given. Set t i := T i -T i-1 , for i ≤ k, with the convention that T 0 = 0. For every i = 1, ..., k, let (L Now we start with the lower bound in [START_REF] Chen | Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes[END_REF]. We use the well known Gram-Hadamard inequality:

(i) t (x) := L T i-1 +t (x) -L T i-1 (x), t ∈ [0, t i ], x ∈ R) be the local time process of B (i) := (B T i-1 +t , t ∈ [0, t i ]). Set D t 1 ,...,t k := det( M t 1 ,...,t k ) with M t 1 ,...,t k := L (i) t i , L (j) t j 1≤i,
D t 1 ,...,t k ≤ k i=1 ||L (i) t i || 2 2 .
By using next the scaling property of Brownian motion, we see that (t

-3/4 i ||L (i) t i || 2 , i ≥ 1) is a sequence of i.i.d. random variables distributed as ||L 1 || 2 . Therefore, E D -1/2 t 1 ,...,t k ≥ c (t 1 . . . t k ) -3/4 , with c := (E[||L 1 || -1 2 ]) k > 0.
We prove now the upper bound in [START_REF] Chen | Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes[END_REF], which is the most difficult part. For this purpose, we introduce the new Gram matrix

M t 1 ,...,t k := t -3 4 i L (i) t i , t -3 4 j L (j) t j i,j
.

Note that all its eigenvalues are nonnegative and denote by λ t 1 ,...,t k the smallest one. We get then

D T 1 ,...,T k = D t 1 ,...,t k = k i=1 t 3/2 i det(M t 1 ,...,t k ) ≥ k i=1 t 3/2 i λ k t 1 ,...,t k .
Thus we can write

C t 1 ,...,t k = E D -1/2 t 1 ,...,t k ≤ (t 1 . . . t k ) -3/4 E λ -k/2 t 1 ,...,t k = (t 1 . . . t k ) -3/4 ∞ 0 P λ -k/2 t 1 ,...,t k ≥ t dt ≤ (t 1 . . . t k ) -3/4 1 + 2 k 1 0 P[λ t 1 ,...,t k ≤ ε] dε ε 1+k/2 .
Therefore Theorem 1 follows from the following proposition: Proposition 8. For any k ≥ 1 and K > 0, there exists a constant C > 0, such that

P(λ t 1 ,...,t k ≤ ε) ≤ C ε K ,
for all ε ∈ (0, 1) and all t 1 , . . . , t k > 0.

Proof of Proposition 8. Note first that

λ t 1 ,...,t k = inf u 2 1 +•••+u 2 k =1 u 1 t -3 4 1 L (1)
t 1 + • • • + u k t -3 4 k L (k) t k 2 2 . ( 8
) Note next that if u 2 1 + • • • + u 2 k = 1, then u max := max i |u i | ≥ 1/ √ k.
Thus dividing all u i by u max leads to

λ t 1 ,...,t k ≥ 1 k min i=1,...,k inf (v j ) j =i , |v j |≤1 t -3 4 i L (i) t i + j =i v j t -3 4 j L (j) t j 2 2
.

Hence, it suffices to bound all terms

P   inf (v j ) j =i , |v j |≤1 t -3 4 i L (i) t i + j =i v j t -3 4 j L (j) t j 2 2 ≤ kε   , (9)
for i ≤ k. By scaling invariance, and changing t j by t j /t i in ( 9), one can always assume that t i = 1. It will also be no loss of generality to assume that i = 1, the case i > 1 being entirely similar. We are thus led to prove that for any k ≥ 1 and K > 0, there exists a constant C > 0, such that for all ε ∈ (0, 1), and all t j > 0,

P   inf (v j ) j>1 , |v j |≤1 L (1) 1 + j≥2 v j t -3/4 j L (j) t j 2 2 ≤ ε   ≤ C ε K . ( 10
)
We want now to bound from below the L 2 -norm by (some power of) the L ∞ -norm using the Hölder regularity of the Brownian local time. To this end, notice that by scaling the constants

C (j) H := sup x =y L (j) t j (x) -L (j) t j (y) t 3/8 j |x -y| 1/4
, for j ≥ 1, are i.i.d. random variables. Moreover, the constant of Hölder continuity of order 1/4 of the j-th term of the sum in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] is larger than or equal to C

(j)

H t -3/8 j
. Since this can be large, we distinguish between indices j such that t j is small from the other ones. More precisely, we define J = {j : t j ≤ ε 4 }, and

E J := ∪ j∈J supp(L (j) t j ),
where supp(f ) denotes the support of a function f . Set also

E ′ J := {x ∈ R : d(x, E J ) < ε} . To simplify notation, set now for all x ∈ R, and v = (v j ) j≥2 , F v (x) := L (1) 1 (x) + j / ∈J, j≥2 v j t -3/4 j L (j) t j (x) and G v (x) := j∈J v j t -3/4 j L (j) t j (x). Notice that G v = 0 on E c J and that sup x =y |F v (x) -F v (y)| |x -y| 1/4 ≤ ε -3/2 j C (j) H . Thus if for some x / ∈ E ′ J , |F v (x)| ≥ ε, and if in the same time j C (j) H ≤ 1/(2ε 1/4 ), then F v + G v 2 2 ≥ x+ε 11 x-ε 11 F v (y) 2 dy ≥ x+ε 11 x-ε 11   ε -ε -3/2 j C (j) H ε 11/4   2 + dy ≥ 1 2 ε 13 .
Moreover, it is known that the C

H have finite moments of any order. Therefore it suffices to prove that for any k ≥ 1 and K > 0, there is a constant C > 0, such that for all ε ∈ (0, 1), and all t 2 , . . . , t k > 0,

P inf v∈R k-1 sup x / ∈E ′ J |F v (x)| ≤ ε ≤ C ε K . (11)
This will follow from the next two lemmas, that we shall prove in the next subsections:

Lemma 9. Let (L t (x), t ≥ 0, x ∈ R) be a continuous in (t,
x) version of the local time process of a standard Brownian motion B. Then for any K > 0 and k ≥ 0, there exist N ≥ 1 and C > 0, such that, for any ε ∈ (0, 1), one can find

N points x 1 , . . . , x N ∈ R, satisfying |x i -x j | ≥ ε 1/8
for all i = j, and

P #{j ≤ N : L 1 (x j ) > ε 1/4 } ≤ k ≤ C ε K . ( 12
)
Lemma 10. For any K > 0 and k ≥ 1, there exist a constant C > 0 and an integer M ≥ 1, such that for all x ∈ R, ε ∈ (0, 1), and t 2 , . . . , t k > 0,

P L (1) 1 (x) > ε 1/4 , inf v∈R k-1 sup |y-x|≤M ε |F v (y)| ≤ ε ≤ C ε K .
Indeed, we can first always assume that E J is included in the union of at most k intervals of length ε, since for any j ∈ J and K ≥ 1, by scaling there exists C > 0, such that

P sup s≤t j B (j) s -B (j) 0 ≥ ε/2 ≤ P sup s≤ε 4 |B s | ≥ ε/2 = P sup s≤1 |B s | ≥ ε -1 /2 ≤ C ε K . ( 13
)
Thus, among any k + 1 points at distance larger than ε 1/8 from each other, at least one of them must be at distance larger than M ε from E ′ J , at least if ε is small enough. Therefore Lemma 9 shows that for any K ≥ 1, there exists C > 0, such that

P inf v sup y / ∈E ′ J |F v (y)| ≤ ε ≤ C ε K + N m=1 P L (1) 1 (x m ) > ε 1/4 , inf v sup |y-xm|≤M ε |F v (y)| ≤ ε ,
where (x 1 , . . . , x N ) are given by Lemma 9. Then (11) follows from the above inequality and Lemma 10. This concludes the proof of Proposition 10.

3.2.

Proof of Lemma 9. We first prove the result for k = 0. Assume without loss of generality that K is an integer larger than 1 and set

X 0 = jε 1/8 : -8K ≤ j ≤ 8K .
Set also s 0 := 0 and for every m ≥ 1,

s m := inf{s > 0 : |B s | ≥ m ε 1/8 }.
Note already that there exists C > 0, such that for all ε ∈ (0, 1),

P(s 8K > 1) ≤ P sup s≤1 |B s | ≤ 8Kε 1/8 ≤ C ε K ,
by using for instance [START_REF] Port | Brownian motion and classical potential theory, Probability and Mathematical Statistics[END_REF]Proposition 8.4 p.52]. Thus it suffices to prove that

P L s 8K (x) ≤ ε 1/4 ∀x ∈ X 0 ≤ ε K , (14)
for all ε ∈ (0, 1). By using the Markov property, and noting that s m ≥ s m-1 + s 1 • θ s m-1 (where θ is the usual shift on the trajectories), we get a.s. for every m ≥ 1,

P L sm (B s m-1 ) -L s m-1 (B s m-1 ) ≤ ε 1/4 | F s m-1 ≤ P(L s 1 (0) ≤ ε 1/4 ).
By the scaling property of the Brownian motion, we know that L s 1 (0) has the same law as ε 1/8 L ′ 1 (0), with L ′ 1 (0) the local time of a standard Brownian motion taken at the first hitting time of {±1}. Moreover, it is known that L ′ 1 (0) is an exponential random variable with parameter 1 (see for instance [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Exercise (4.12) chap VI, p. 265). Therefore, a.s. for every m ≥ 1,

P L sm (B s m-1 ) -L s m-1 (B s m-1 ) ≤ ε 1/4 | F s m-1 ≤ P(L ′ 1 (0) ≤ ε 1/8 ) ≤ ε 1/8
. Then we get by induction,

P L s 8K (x) ≤ ε 1/4 ∀x ∈ X 0 ≤ P L sm (B s m-1 ) -L s m-1 (B s m-1 ) ≤ ε 1/4 ∀m ≤ 8K ≤ ε K ,
proving [START_REF] Doussal | Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields[END_REF]. This concludes the proof of the lemma for k = 0. Now we prove the result for general k ≥ 0. For m ∈ Z, consider the set

X m := m(16K + 1)ε 1/8 + X 0 .
Then the proof above shows similarly that for any 0 ≤ m ≤ k,

P L 1 (x) ≤ ε 1/4 ∀x ∈ X m ∪ X -m ≤ C ε K .
The lemma follows immediately.

3.3.

Proof of Lemma 10. Let K > 0 be fixed, and assume without loss of generality that x ≥ 0. Fix also M ≥ 1 some integer to be chosen later.

For every affine subspace V of R M , we denote by V ε the set

V ε := {v ∈ R M : d(v, V ) ≤ ε}, where d(v, V ) = min{|v -y| ∞ : y ∈ V }. Then we can write P L (1) 1 (x) > ε 1/4 , inf v∈R k-1 sup |y-x|≤M ε |F v (y)| ≤ ε ≤ P L (1) 1 (x) > ε 1/4 , (L (1) 1 (x + ε), . . . , L (1) 1 (x + M ε)) ∈ V ε := P ε , where V := V ect L (j) t j (x + ℓε) ℓ=1,...,M , j ∈ I , with I := {j > 1 : j / ∈ J}. Set now τ := inf{s > 0 : L (1) s (x) > ε 1/4 }, and for y ≥ 0, Y (y) := L (1)
τ (x + y). It follows from the second Ray-Knight theorem (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], Theorem (2.3) p.456) that Y is equal in law to a squared Bessel process of dimension 0 starting from ε 1/4 . Moreover, with this notation, we can write

P ε = P [τ < 1 and (Y (ε), . . . , Y (M ε)) ∈ V * ε ] , (15)
with

V * := (Y (ℓε) -L (1)
1 (x + ℓε)) ℓ=1,...,M + V, which is an affine space of R M , of dimension at most k -1 .

Observe now that even on the event {τ < 1}, the space V * is not independent of Y and τ , since its law depends a priori on τ . However, if this was true (and we will see below how one can reduce the proof to this situation), then P ε would be dominated by

sup V P [(Y (ε), . . . , Y (M ε)) ∈ V ε ] ,
with the sup taken over all affine subspaces V ⊆ R M of dimension at most k -1. This last term in turn is controlled by the following lemma, whose proof is postponed to the next subsections.

Lemma 11. Let Y be a squared Bessel process of dimension 0 starting from ε 1/4 . For any M ≥ 1 and k ≥ 1, there exists C > 0, such that for all ε ∈ (0, 1),

sup V P [(Y (ε), . . . , Y (M ε)) ∈ V ε ] ≤ C ε (5M -4(k-1))/8 , ( 16
)
where the sup is over all affine subspaces V ⊆ R M of dimension at most k -1.

So at this point we are just led to see how one can solve the problem of the dependence between V * and τ . To this end, we introduce the time τ ′ spent by B (1) above x before time τ , which by the occupation times formula (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], Theorem (2.3) p.456) is equal to:

τ ′ := τ 0 1 {B (1) s ≥x} ds = ∞ 0 Y (y) dy.
Moreover, τ ′ is also equal in law to the first hitting time of ε 1/4 /2 by a Brownian motion (see the proof of Theorem (2.7) p.243 in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]). In particular

P(τ ′ ≤ ε 3/4 ) = O(ε K ). ( 17
)
Next instead of using Lemma 11, we will need the following refinement:

Lemma 12. Let M ≥ 1 be some integer. Let Y be a squared Bessel process of dimension 0 starting from ε 1/4 . Set

A ε := |Y (M ε) -ε 1/4 | ≤ ε 1/2 and M ε 0 Y (y) dy < ε . Then P(A c ε ) = O(ε K ).
Moreover, for any M ≥ 1 and k ≥ 1, there exists C > 0, such that for any affine space V of dimension at most k -1, almost surely for all ε ∈ (0, 1),

1 { ∞ 0 Y (y) dy≥ε 3/4 } P (Y (ε), . . . , Y (M ε)) ∈ V ε , A ε ∞ 0 Y (y) dy ≤ C ε (5M -4(k-1))/8 .
We postpone the proof of this lemma to the next subsections, and we conclude now the proof of Lemma 10. First it follows from the excursion theory of the Brownian motion that, conditionally to τ ′ , Y is independent of τ . On the other hand, conditionally to τ , V * is independent of τ ′ and Y . Let M be an integer such that M ≥ (4(k -1) + 8K)/5. According to [START_REF] Gall | processus de branchement et superprocessus[END_REF], [START_REF] Port | Brownian motion and classical potential theory, Probability and Mathematical Statistics[END_REF] and to the first part of Lemma 12, we get

P ε ≤ P τ < 1, τ ′ ≥ ε 3/4 , (Y (ε), . . . , Y (M ε)) ∈ V * ε , A ε + O(ε K ) ≤ E 1 {τ <1,τ ′ ≥ε 3/4 } E[f (Y, V * )|τ, τ ′ ] + O(ε K ), with f (y, V ) := 1 {(y(ε),...,y(M ε))∈Vε}∩{|y(M ε)-ε 1/4 |≤ε 1/2
and M ε 0 y(s) ds<ε} . Now, since on one hand Y and V * are independent conditionally to (τ, τ ′ ), and on the other hand Y and τ are independent conditionally to τ ′ , we have

E[f (Y, V * )|τ, τ ′ ] = E[f (Y, V )|τ, τ ′ ] dP V * |(τ,τ ′ ) (V ) = E[f (Y, V )|τ ′ ] dP V * |(τ,τ ′ ) (V ).
Since moreover, V * and τ ′ are independent conditionally to τ , we get

E[f (Y, V * )|τ, τ ′ ] = E[f (Y, V )|τ ′ ] dP V * |τ (V ). (18)
Hence, according to our choice of M ,

P ε ≤ E 1 {τ <1,τ ′ ≥ε 3/4 } E[f (Y, V )|τ ′ ] dP V * |τ (V ) + O(ε K ) ≤ E 1 {τ <1,τ ′ ≥ε 3/4 } 1 b(V,ε,τ ′ ) dP V * |τ (V ) + O(ε K ), with b(V, ε, t ′ ) := 1 {τ ′ ≥ε 3/4 } E[f (Y, V )|τ ′ = t ′ ] > C ε (5M -4(k-1))/8 .
The second part of Lemma 12 insures that, for every affine subspace V of dimension at most k -1 of R M , we have

1 b(V,ε,t ′ ) = 0 for P τ ′ -almost every t ′ > 0.
However, since b(V, ε, τ ′ ) depends a priori on V , we cannot conclude directly. But it is well known that τ ′ admits a positive density function on (0, +∞) (see (24) below for an explicit expression). Therefore, for every V ,

1 b(V,ε,t ′ ) = 0 for Lebesgue almost every t ′ > 0. ( 19
)
Now it follows from the excursion theory that τ ′ and ττ ′ are independent and identically distributed. Therefore (τ, τ ′ ) admits a continuous density function h on (0, +∞) 2 and we have

P ε ≤ O(ε K ) + 1 0 t 0 1 b(V,ε,t ′ ) dP V * |τ =t (V ) h(t, t ′ ) dt ′ dt ≤ O(ε K ) + 1 0 t 0 1 b(V,ε,t ′ ) h(t, t ′ ) dt ′ dP V * |τ =t (V ) dt (20) ≤ O(ε K ),
the last term of (20) being equal to zero according to [START_REF] Xiao | The Hausdorff dimension of the level sets of stable processes in random scenery[END_REF]. This concludes the proof of Lemma 10.

It remains now to prove Lemma 12. Its proof uses Lemma 11, so let us start with the proof of the latter.

3.4.

Proof of Lemma 11. We first prove the following result:

Lemma 13. For every K > 0 and M ≥ 1, there exists C > 0, such that for all ε ∈ (0, 1),

P ∃ℓ ∈ {1, . . . , M } : |Y (ℓε) -ε 1/4 | > ε 1/2 ≤ C ε K .
Proof. Recall that Y is solution of the stochastic differential equation

Y (y) = ε 1/4 + 2 y 0 Y (u) dβ u for all y ≥ 0,
where β is a Brownian motion (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Ch. XI). In particular, Y is stochastically dominated by the square of a one-dimensional Brownian motion starting from ε 1/8 . Then it follows that, for some constant C > 0, whose value may change from line to line, but depending only on K and M ,

P ∃ℓ ∈ {1, . . . , M } : |Y (lε) -ε 1/4 | > ε 1/2 ≤ P sup s≤M ε |Y (s) -ε 1/4 | > ε 1/2 ≤ C ε -4K E M ε 0 Y (u) du 4K
by the Burkholder-Davis-Gundy inequality,

≤ C ε -1 M ε 0 E Y (u) 4K du ≤ C ε -1 M ε 0 E (ε 1/8 + B u ) 8K du ≤ C ε K ,
with B some standard Brownian motion. This concludes the proof of the lemma.

We continue now the proof of Lemma 11. Set

B ∞ (ε 1/4 , ε 1/2 ) := (y 1 , . . . , y M ) ∈ R M : y ℓ -ε 1/4 ≤ ε 1/2 ∀ℓ ∈ {1, . . . , M } .
Lemma 13 shows that for any V of dimension at most k -1,

P [(Y (ε), . . . , Y (M ε)) ∈ V ε ] ≤ P (Y (ε), . . . , Y (M ε)) ∈ B ∞ (ε 1/4 , ε 1/2 ) ∩ V ε + C ε K . ( 21
) Next observe that B ∞ (ε 1/4 , ε 1/2 )∩ V ε , can be covered by O(ε -(k-1)/2
) balls of radius ε. It follows that

P (Y (ε), . . . , Y (M ε)) ∈ B ∞ (ε 1/4 , ε 1/2 ) ∩ V ε ≤ C ε -(k-1)/2 sup x∈B∞(ε 1/4 ,ε 1/2 ) P [(Y (ε), . . . , Y (M ε)) ∈ B ∞ (x, ε)] . (22)
Now for y > 0, denote by Y y a squared Bessel process with dimension 0 starting from y. An explicit expression of its semigroup is given just after Corollary (1.4) p.441 in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]. In particular when y > ε 1/4 /2, the law of Y y (ε) is the sum of a Dirac mass at 0 with some negligible weight and of a measure with density

z → q ε (y, z) := (2ε) -1 y z exp - y + z 2ε I 1 √ yz ε ,
where I 1 is the modified Bessel function of index 1. Moreover it is known (see (5.10.22) or (5.11.10) in [START_REF] Lebedev | Special functions and their applications[END_REF]), that

I 1 (z) = O(e z / √ z), as z → ∞. Thus sup |y-ε 1/4 |≤ε 1/2 sup |z-ε 1/4 |≤ε 1/2 q ε (y, z) = O(ε -3/8 ). It follows that sup |x-ε 1/4 |≤ε 1/2 sup |y-ε 1/4 |≤ε 1/2 P [|Y y (ε) -x| ≤ ε] = O(ε 5/8 ).
Then by using the Markov property and Lemma 13, we get by induction

sup x∈B∞(ε 1/4 ,ε 1/2 ) P [(Y (ε), . . . , Y (M ε)) ∈ B ∞ (x, ε)] ≤ C ε 5M/8 . ( 23
)
Since all the constants in our estimates are uniform in V , Lemma 11 follows from (21), ( 22) and (23).

3.5. Proof of Lemma 12. Let K > 0 be given. Lemma 13 shows in particular that

P |Y (M ε) -ε 1/4 | > ε 1/2 = O(ε K ).
Next, recall that Y is stochastically dominated by the square of a one-dimensional Brownian motion starting from ε 1/8 . It follows that

P M ε 0 Y (y) dy ≥ ε = O(ε K ),
and this already proves the first part of the lemma.

It remains to prove the second part. We deduce it from Lemma 11. To simplify notation, from now on we will denote the integral of Y on [0, ∞) by

∞ 0 Y . Likewise M ε 0 Y and ∞ M ε Y will have analogous meanings. For any affine subspace V ⊆ R M of dimension at most k -1, set A ′ ε (V ) := {(Y (ε), . . . , Y (M ε)) ∈ V ε } ∩ A ε .
Then for any nonnegative bounded measurable function φ supported on [ε 3/4 , ∞), we can write

E φ ∞ 0 Y P A ′ ε (V ) ∞ 0 Y = E φ ∞ 0 Y , A ′ ε (V ) = E φ M ε 0 Y + ∞ M ε Y , A ′ ε (V ) .
Now we recall that if Y y denotes a squared Bessel process of dimension 0 starting from some y > 0, then ∞ 0 Y y is equal in law to the first hitting time of y/2 by some Brownian motion, and thus has density given by

f y (t) := y 2 (2πt 3 ) -1/2 exp(-(y/2) 2 /2t
) for all t > 0 and y > 0,

see for instance [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] p.107. In particular

sup t≥ε 3/4 sup t ′ ≤ε sup |y-ε 1/4 |≤ε 1/2 f y (t -t ′ ) f ε 1/4 (t) < ∞.
Then by using the Markov property and Lemma 11, we get

E φ ∞ 0 Y P A ′ ε (V ) ∞ 0 Y = E ∞ ε 3/4 φ(t) f Y (M ε) t - M ε 0 Y dt, A ′ ε (V ) ≤ C P A ′ ε (V ) E φ ∞ 0 Y ≤ C ε (5M -4(k-1))/8 E φ ∞ 0 Y .
Since this holds for any φ, this proves the second part of Lemma 12, as wanted.

Proof of Theorem 3 and Corollary 4

We start with the proof of Theorem 3. We follow the general strategy which is used in the case of the Brownian motion, as for instance in Le Gall's course [START_REF] Gall | processus de branchement et superprocessus[END_REF]Chapter 2].

Consider the regularizing function

p ε (y) := 1 √ 2πε exp - y 2 2ε ε > 0 y ∈ R,
and recall that by Fourier inversion

p ε (y) = 1 2π R exp(iyξ - 1 2 ε|ξ| 2 ) dξ.
Define then for all ε ∈ (0, 1], t > 0, and x ∈ R,

L(ε, t, x) := t 0 p ε (∆ s -x) ds.
As explained in [START_REF] Gall | processus de branchement et superprocessus[END_REF], it suffices to control the three terms:

E L(ε, t, x) -L(ε, t, x ′ ) 2p , E L(ε, t, x) -L(ε ′ , t, x) 2p , E L(ε, t, x) -L(ε, t ′ , x) 2p .
For the first term, some elementary computation shows that

E L(ε, t, x) -L(ε, t, x ′ ) 2p ≤ c p R 2p dξ 1 . . . dξ 2p 2p j=1 e -ixξ j -e -ix ′ ξ j Ξp ds 1 . . . ds 2p E e i ξ j ∆s j , (25)
with c p some positive constant (whose value may change in the following lines) and

Ξ p := {s 1 ≤ • • • ≤ s 2p ≤ t}.
We use next that for any γ ∈ (0, 1], and any y, y ′ ∈ R,

e iy -e iy ′ ≤ c |y -y ′ | γ ,
for some constant c > 0. Moreover, if η j = ξ j + • • • + ξ 2p , and t j = s js j-1 , for all j ≥ 1 (with the convention s 0 = 0), then

E e i ξ j ∆s j = E e -1 2 i,j η i η j L (i) t i ,L (j) t j
= E e -1 2 Mt 1 ,...,t 2p η,η , with η = (η 1 , . . . , η 2p ). Therefore a change of variables in (25) gives ). Then we fix some T > 0, and by using also that for all j, and t ≤ T ,

E L(ε, t, x) -L(ε, t, x ′ ) 2p ≤ c p |x -x ′ | 2γp R 2p dη 2p j=1 |η j+1 -η j | γ [0,t] 2p dt 1 . . . dt 2p E e -
|t -3/4 j+1 η j+1 -t -3/4 j η j | ≤ c max(t -3/4 j+1 , t -3/4 j ) |η| ≤ c t -3/4 j+1 t -3/4 j T 3/4 |η|,
for some constant c > 0, we get for all t ≤ T ,

E L(ε, t, x) -L(ε, t, x ′ ) 2p ≤ c p,T |x -x ′ | 2γp [0,t] 2p dt 1 . . . dt 2p   2p j=1 t -3/4(1+2γ) j   R 2p
dη E e -

E L(ε, t, x) -L(ε, t, x ′ ) 2p ≤ c p,T |x -x ′ | 2γp ,
for all t ≤ T .

A similar computation yields to an analogous estimate for the second term, except that this time we need to choose γ < 1/12: for all p ≥ 1, T > 0, and γ < 1/12, there exists some constant c ′ p,T > 0, such that for all x ∈ R, all ε, ε ′ > 0, and all t ≤ T ,

E L(ε, t, x) -L(ε ′ , t, x) 2p ≤ c ′ p,T ε -ε ′ 2pγ .
Now the estimate of the last term is easier. After some calculation and by using Theorem 1, we get for t < t ′ ,

E L(ε, t, x) -L(ε, t ′ , x) 2p ≤ c p t≤s 1 ≤•••≤s 2p ≤t ′ ds 1 . . . ds 2p (s j -s j-1 ) 3/4 , which shows that E L(ε, t, x) -L(ε, t ′ , x) 2p ≤ c p |t ′ -t| p/2 .
Then Part (i) and (ii) in Theorem 3 follow from Kolmogorov's criterion (see [START_REF] Gall | processus de branchement et superprocessus[END_REF] for details). For (iii), first observe that (ii) implies that a.s. for any t > 0 and x ∈ R,

L t (x) = lim ε→0 1 2ε t 0 1 {∆s∈[x-ε,x+ε]} ds.
Then (iii) immediately follows from this equation and the property of self similarity of ∆. For (iv), we can observe that by using the above computations and the dominated convergence theorem, we get

E[L t (x) k ] = lim ε→0 E[L(ε, t, x) k ].
Part (iv) follows. Part (v) is immediate, and was already observed in [START_REF] Xiao | The Hausdorff dimension of the level sets of stable processes in random scenery[END_REF].

Concerning Corollary 4, the upper bound was already proved in [START_REF] Xiao | The Hausdorff dimension of the level sets of stable processes in random scenery[END_REF] and [START_REF] Khoshnevisan | The codimension of the zeros of a stable process in random scenery[END_REF] and was considered there as the easiest part. So we only care about the lower bound here. For this we can use Frostman's Lemma together with Theorem 3, which directly proves the result (see [START_REF] Gall | processus de branchement et superprocessus[END_REF] for instance).

Proof of Theorem 5

In most of this section, t 1 , . . . , t k , are fixed positive reals. Moreover, by convention a function f (n 1 , . . . , n k ) is said to be a o k (g(n)), for some function g, if it converges to 0 after multiplication by 1/g(n), when n → ∞ and n i /n → t i for all i ≥ 1. Analogous convention is used for the notation O k (g(n)).

Recall that (S m , m ≥ 0) denotes the random walk. For every i = 1, ..., k, let (N

(i) m (x), 1 ≤ m ≤ n i , x ∈ Z), be the local time process of S (i) m := S n 1 +...+n i-1 +m , 0 ≤ m ≤ n i -1 .
In other words,

N (i) m (x) := #{k = 0, ..., m -1 : S n 1 +...+n i-1 +k = x} = N n 1 +...+n i-1 +m (x) -N n 1 +...+n i-1 (x),
for all i ≤ k. Set also

D n 1 ,...,n k := det N (i) n i , N (j) n j 1≤i,j≤k
, where here •, • denotes the usual scalar product on ℓ 2 (Z).

5.1.

Inverse Fourier transform and a periodicity issue. The first step in local limit theorems is often the use of Fourier inverse transform. This is essentially the content of the next lemma. Before stating it, let us introduce some new notation. Recall that ϕ ξ denotes the characteristic function of ξ 0 . Let now ϕ n 1 ,...,n k be the characteristic function of (Z

n 1 +•••+n i - Z n 1 +•••+n i-1 ) i=1,.
..,k . Since (ξ y ) y∈Z is a sequence of i.i.d. random variables, which is independent of S, we have for all (θ 1 , . . . ,

θ k ) ∈ R k , ϕ n 1 ,...,n k (θ 1 , . . . , θ k ) := E   y∈Z ϕ ξ   k j=1 θ j (N n 1 +...+n j (y) -N n 1 +...+n j-1 (y))     = E   y∈Z ϕ ξ   k j=1 θ j N (j) n j (y)     . ( 26
)
We can now state the announced lemma.

Lemma 14. If n i ∈ d 0 Z for all i ≤ k, then P (Z n 1 = • • • = Z n 1 +•••+n k = 0) = d 2π k [-π d , π d ] k ϕ n 1 ,...,n k (θ 1 , . . . , θ k ) dθ 1 . . . dθ k . Otherwise P(Z n 1 = • • • = Z n 1 +•••+n k = 0) = 0.
Proof. Since Z is Z-valued, we immediately get

P(Z n 1 = • • • = Z n 1 +•••+n k = 0) = P(Z n 1 = • • • = Z n 1 +•••+n k -Z n 1 +•••+n k-1 = 0) = 1 (2π) k [-π,π] k ϕ n 1 ,...,n k (θ 1 , . . . , θ k ) dθ 1 . . . dθ k .
Notice now that e 2iπξ 0 /d = ϕ ξ (2π/d) almost surely and that ϕ ξ (2π/d) d = 1. Hence, for any integer m ≥ 0 and any u ∈ R,

ϕ ξ (2mπ/d + u) = ϕ ξ (2π/d) m ϕ ξ (u).
We deduce that, for every (l 1 , . . . , l k ) ∈ Z k , we have

ϕ n 1 ,...,n k θ 1 + 2l 1 π d , . . . , θ k + 2l k π d = E   y∈Z ϕ ξ   k j=1 θ j + 2l j π d N (j) n j (y)     = E   y∈Z ϕ ξ (2π/d) k j=1 l j N (j) n j (y) ϕ ξ   k j=1 θ j N (j) n j (y)     = ϕ ξ (2π/d) k j=1 l j n j ϕ n 1 ,...,n k (θ 1 , . . . , θ k ), since y N (j) n j (y) = n j . But, if n j ∈ d 0 Z for all j ≤ k, then ϕ ξ (2π/d) k j=1 n j l j = 1
, for all (l 1 , . . . , l k ) ∈ Z k , and the result follows with a change of variables. If not, let j be such that n j / ∈ d 0 Z. Then ϕ ξ (2π/d) n j is a nontrivial d-th root of unity and we can write

P(Z n 1 = . . . = Z n 1 +•••+n k = 0) = 1 (2π) k   d-1 l j =0 ϕ ξ (2π/d) n j l j   × [-π,π] k-1 -[ π d , π d ] ϕ n 1 ,...,n k (θ 1 , . . . , θ k ) dθ j dθ 1 . . . dθ j-1 dθ j+1 . . . dθ k = 0.
This concludes the proof of the lemma.

5.2.

A typical behaviour for random walks. We want to argue that typically the simple random walk visits roughly √ n sites before time n; spends time of order at most √ n on each of them, and that its local time process is Hölder continuous of order 1/2, with a Hölder constant in O(n 1/4 ). This is true with high probability if we allow some correction of order n γ , with γ > 0. This is the content of the next lemma, which can be proved as Lemma 6 in [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF] and is standard. Set for all i ≤ k, N * i := sup y N (i) n i (y) and R i := #{y :

N (i) n i (y) > 0}.
Lemma 15. For every n ≥ 1 and γ > 0, set Ω n 1 ,...,n k := Ω

(1)

n 1 ,...,n k ∩ Ω (2)
n 1 ,...,n k , where

Ω (1) n 1 ,...,n k := R i ≤ n 1 2 +γ i ∀i ≤ k ,
and

Ω (2) n 1 ,...,n k := sup y =z |N (i) n i (y) -N (i) n i (z)| |y -z| 1/2 ≤ n 1 4 +γ i ∀i ≤ k .
Then, for every p,

P(Ω c n 1 ,...,n k ) = o(min i n -p i ).
Note that on Ω n 1 ,...,n k , for every i, we have

N * i ≤ n 1 2 +γ i and V (i) n i := y (N (i) n i (y)) 2 ≤ n 3 2 +3γ i . 5.
3. Scheme of the proof. We follow roughly the same lines as for the proof of Theorem 1 in [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF]. However the situation is more complicated here, since we consider multiple times in a non-markovian context. Moreover, we want upper bounds which are uniform in n 1 , . . . , n k , and this also requires some additional care.

First we have to see that the main contribution in the estimate comes from the integral near the origin. Recall in particular the notation from (26).

Proposition 16. Let η ∈ (0, 1/8) be given. Then, for every t 1 , . . . , t k ∈ (0, 1), we have

U (η) ϕ n 1 ,...,n k (θ 1 , . . . , θ k ) dθ 1 . . . dθ k = √ 2π σ k C t 1 ,...,t k n -3k/4 + o k (n -3k/4 ),
where

U (η) := {|θ i | ≤ n -1 2 -η i ∀i ≤ k}. Moreover, for every θ ∈ (0, 1), sup n≥1 sup n θ ≤n 1 ,...,n k ≤n k i=1 n 3 4 i U (η) ϕ n 1 ,...,n k (θ 1 , . . . , θ k ) dθ 1 . . . dθ k < ∞.
The next two propositions show that the rest of the integral is negligible.

Proposition 17. Let η ∈ (0, 1/8) be given. Then, for every t 1 , . . . , t k ∈ (0, 1), we have

V (η) |ϕ n 1 ,...,n k (θ 1 , . . . , θ k )| dθ 1 . . . dθ k = o k (n -3k/4 ) where V (η) := {|θ i | ≤ n -1 2 +η i ∀i ≤ k} ∩ {∃j : |θ j | ≥ n -1 2 -η j }. Moreover, for every θ ∈ (0, 1), sup n≥1 sup n θ ≤n 1 ,...,n k ≤n k i=1 n 3 4 i V (η) |ϕ n 1 ,...,n k (θ 1 , . . . , θ k )| dθ 1 . . . dθ k < ∞.
Proposition 18. Let η ∈ (0, 1/2) and θ ∈ (0, 1) be given. Then there exists c > 0 such that

sup n≥1 sup n θ ≤n 1 ,...,n k ≤n {∃i : |θ i |>n -1 2 +η i } |ϕ n 1 ,...,n k (θ 1 , . . . , θ k )| dθ 1 . . . dθ k = o(e -n c ).
This last proposition can be proved by using exactly the same argument as in the proof of Proposition 10 in [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF]. The only difference is that, if say |θ i | > n -1/2+η i , then after having defined peaks for S (i) , we need to work also conditionally to all N (j) n j , for j = i. But this does not change anything to the proof. Since it would be fastidious to reproduce the argument, we will not prove this proposition here and we refer the reader to [START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF] for details.

Note that Theorem 5 readily follows from these propositions and Lemma 14.

5.4.

Proof of Proposition 16. We will use Borodin's result [START_REF] Borodin | On the character of convergence to Brownian local time. II[END_REF] on approximations of Brownian local time by random walks local time. He proved in particular (see Remark 1.3 in [START_REF] Borodin | On the character of convergence to Brownian local time. II[END_REF]) that under some moment condition on the random walk, and on a suitable probability space, for all T > 0 and all γ > 0, there exist constants C > 0 and δ > 0, such that for all n ≥ 1,

P (E c n ) ≤ C n -1-δ , with E n := sup (t,x)∈[0,T ]×R |N [nt] ([ √ nx]) - √ nL t (x)| ≤ Cn 1 4 ln n, B 1 - S n √ n ≤ n -1 4 +γ ,
where N and L are the local time processes, respectively of the random walk S and of the Brownian motion B. But a careful look at his proof shows actually that if the random walk increments have finite moments of any order, then the above holds for any δ > 0 (see Formulas (3.8) and (3.9) and Lemma 3.2).

By using now this result, Lemma 15 and the Markov property of the random walk and of Brownian motion, we deduce the following:

Lemma 19. Let γ ∈ (0, 1/4) and k ≥ 1 be given. Then for every n ≥ 1 and every 0 ≤ n 1 , ..., n k ≤ n, it is possible to construct the Brownian motion and the random walk on a suitable probability space, such that for all p > 0,

P F c n,n 1 ,...,n k = O (min i n i ) -p , where F n,n 1 ,...,n k = F 1 (n, n 1 , . . . , n k ) ∩ • • • ∩ F 4 (n, n 1 , .
. . , n k ), and (with

t i = n i /n for i ≤ k), F 1 (n, n 1 , . . . , n k ) := sup x∈R N (i) n i ( √ nx) - √ nL (i) t i (x) ≤ n 1 4 +γ i ∀i ≤ k , F 2 (n, n 1 , . . . , n k ) := sup{|x -S (i) 0 | : N (i) n i (x) = 0} ≤ t 1/2 i n 1 2 +γ ∀i ≤ k , F 3 (n, n 1 , . . . , n k ) := sup{|x -B (i) 0 | : L (i) t i (x) = 0} ≤ t 1/2 i n γ ∀i ≤ k , F 4 (n, n 1 , . . . , n k ) := sup k N (i) n i (k) ≤ t 1/2 i n 1 2 +γ and sup x L (i) t i (x) ≤ t 1/2 i n γ ∀i ≤ k .
The proof of this result is elementary and left to the reader. Define now for all ε > 0, the set

Ω n 1 ,...,n k (ε) := k i=1 n -3 2 i D n 1 ,...,n k ≥ ε . (27)
We then obtain the following:

Lemma 20. Let θ ∈ (0, 1) and θ 0 ∈ (0, θ/4) be given. Then for every L > 0, we have

sup n≥1 sup n θ ≤n 1 ,...,n k ≤n sup ε≥n -θ 0 ε -L P Ω c n 1 ,...,n k (ε) < ∞,
and for every p > 0,

sup n≥1 sup n θ ≤n 1 ,...,n k ≤n E k i=1 n 3p 2 i D -p n 1 ,...,n k , Ω n 1 ,...,n k (n -θ 0 ) < ∞.
Proof. Let γ > 0 be such that θ 0 < (θ/4) -3γk and let L > 0 be fixed. Thanks to the previous lemma we can assume that the Brownian motion B and the random walk S are constructed on a space, where P(F c n,n 1 ,...,n k ) = O(n -p ), for all p > 0. Now for all i, j, set

A (n) i,j := (n i n j ) -3/4 y N (i)
n i (y)N (j) n j (y) and A i,j := (t i t j ) -3/4 R L (i)

t i (x)L (j)
t j (x) dx, with t i = n i /n and t j = n j /n. First, we rewrite A (n) i,j as follows

A (n) i,j = (t i t j ) -3 4 R N (i) n i (⌊ √ nx⌋) √ n N (j) n j (⌊ √ nx⌋) √ n dx.
Observe next that, on F n,n 1 ,...,n k , for all i, j and n i , n j ≤ n, we have

A (n) i,i ≤ n 3γ , A i,i ≤ n 3γ , (28) and t
-3 2 i R N (i) n i (⌊ √ nx⌋) √ n -L (i) t i (x) 2 dx ≤ t -3 2 i 2t 1 2 i n γ t 1 2 i n -1 2 +2γ ≤ 2t -1 2 i n -1 2 +3γ .
Hence, with the use of the Cauchy-Schwartz inequality, we get

A (n) i,j ≤ n 3γ , A i,j ≤ n 3γ and A (n) i,j -A i,j ≤ 2 √ 2t -1 4 i n -1 4 +3γ ≤ 4n -θ 4 +3γ . (29)
We use next that

k i=1 n -3 2 i D n 1 ,...,n k = det (A (n)
i,j ) i,j and det M t 1 ,...,t k = det ((A i,j ) i,j ) .

Furthermore, for any matrix M :

det

((M i,j ) i,j ) = σ∈S k (-1) sgn(σ) k i=1 M i,σ(i) ,
where S k is the group of permutations of {1, ..., k} and sgn(σ) is the signature of σ. Therefore, using (29), on F n,n 1 ,...,n k , when n i ≤ n, for all i ≤ k, we get for n large enough,

k i=1 n -3 2 i D n 1 ,...,n k -det M t 1 ,...,t k ≤ σ∈S k k i=1 n 3γ(k-1) A (n) i,σ(i) -A i,σ(i) ≤ 4 (k + 1)! n 3γk n -θ 4 ≤ n -θ 0 ,
according to our assumption on γ.

Thus, for n large enough, and every ε ≥ n -θ 0 , we get by using Proposition 8

1 k = O ε L ,
with λ t 1 ,...,t k as in Proposition 8. So we just proved that for any L > 0, the constant

C L := sup n≥1 sup ε≥n -θ 0 ε -L sup n θ ≤n 1 ,...,n k ≤n P Ω c n 1 ,...,n k (ε) ,
is finite, which gives the first part of the lemma. Then we get for any p > 0, sup

n θ ≤n 1 ,...,n k ≤n E k i=1 n 3p 2 i D -p n 1 ,...,n k , Ω n 1 ,...,n k (n -θ 0 ) = sup n θ ≤n 1 ,...,n k ≤n ∞ 0 P n -θ 0 ≤ k i=1 n -3 2 i D n 1 ,...,n k ≤ t -1/p dt = sup n θ ≤n 1 ,...,n k ≤n p +∞ n -θ 0 P n -θ 0 ≤ k i=1 n -3 2 i D n 1 ,...,n k ≤ ε dε ε p+1 ≤ p 1 n -θ 0 C p+1 dε + p +∞ 1 dε ε p+1 < ∞,
where for the third line we have used the change of variables t = ε -p . This concludes the proof of the lemma.

The next step is the Lemma 21. Let η ∈ (0, 1/4) and θ ∈ (0, 1) be given. Then

lim n→∞ sup n θ ≤n 1 ,...,n k ≤n i n 3/4 i × U (η)
ϕ n 1 ,...,n k (θ 1 , . . . , θ k ) -E e -σ 2 Qn 1 ,...,n k (θ 1 ,...,θ k )/2 dθ 1 . . . dθ k = 0, where

Q n 1 ,...,n k (θ 1 , . . . , θ k ) := y θ 1 N (1) n 1 (y) + • • • + θ k N (k) n k (y) 2 . Proof. Recall that U (η) = {|θ i | ≤ n -1 2 -η i ∀i ≤ k}. Set E n 1 ,...,n k (θ 1 , . . . , θ k ) := y ϕ ξ (θ 1 N (1) n 1 (y) + • • • + θ k N (k) n k (y)
)e -σ 2 Qn 1 ,...,n k (θ 1 ,...,θ k )/2 .

We have to prove that

U (η) E [|E n 1 ,...,n k (θ 1 , . . . , θ k )|] dθ 1 . . . dθ k = o k i=1 n -3/4 i .
Observe that

E n 1 ,...,n k (θ 1 , . . . , θ k ) = y z>y exp - σ 2 2 (θ 1 N (1) n 1 (z) + • • • + θ k N (k) n k (z)) 2 × ϕ ξ (θ 1 N (1) n 1 (y) + • • • + θ k N (k) n k (y)) -e -σ 2 2 (θ 1 N (1) n 1 (y)+•••+θ k N (k) n k (y)) 2 × z<y ϕ ξ θ 1 N (1) n 1 (z) + • • • + θ k N (k) n k (z) .
Recall now that, since ξ is square integrable, we have 1ϕ ξ (u) ∼ σ 2 |u| 2 /2, as u → 0. It follows that,

ϕ ξ (u) -e -σ 2 u 2 /2 ≤ |u| 2 h(|u|) for all u ∈ R,
with h a continuous and monotone function on [0, +∞) vanishing in 0. In particular there exists a constant ε 0 > 0, such that

|ϕ ξ (u)| ≤ exp -σ 2 |u| 2 /4 for all u ∈ [-ε 0 , ε 0 ]. (30)
Fix now γ ∈ (0, η) and θ 0 ∈ (0, θ/4). Next recall (27) and observe that on

Ω(γ, θ 0 ) := Ω n 1 ,...,n k ∩ Ω n 1 ,...,n k (n -θ 0 ), if |θ i | ≤ n -1 2 -η i
for all i ≤ k, then (see the remark following Lemma 15) for all z ∈ Z,

|θ 1 N (1) n 1 (z) + • • • + θ k N (k) n k (z)| ≤ kn γ-η
, which is smaller than ε 0 for n large enough. Then we get, Therefore a change of variables gives Finally Proposition 16 is deduced from the following lemma.

|E n 1 ,...,n k (θ 1 , . . . , θ k )|1 Ω(γ,θ 0 ) ≤ h(kn γ-η )e -σ 2 Qn 1 ,...,n k (θ 1 ,...,θ k )/4 × y e σ 2 4 (θ 1 N (1) n 1 (y)+•••+θ k N (k) n k (y)) 2 θ 1 N (1) n 1 (y) + • • • + θ k N (k) n k (y) 2 = o(1) × e (σε 0 ) 2 e -
U (η) |E n 1 ,...,n k (θ 1 , . . . , θ k )| 1 Ω(γ,θ 0 ) dθ 1 . . . dθ k = o(1) × D -1/2 n 1 ,...,n k R k e -σ 2
Lemma 22. Let t 1 , . . . , t k ∈ (0, 1) and η ∈ (0, 1/8) be given. Then

U (η) E e -σ 2 Qn 1 ,...,n k (θ 1 ,...,θ k )/2 dθ 1 . . . dθ k = √ 2π σ k C t 1 ,...,t k n -3k/4 + o k (n -3k/4 ).
Proof. First write

U (η)
e -σ 2 Qn 1 ,...,n k (θ 1 ,...,θ k )/2 dθ 1 . . . dθ k = I n 1 ,...,n k -J n 1 ,...,n k , where I n 1 ,...,n k is the integral over R k and J n 1 ,...,n k is the integral over {∃j :

|θ j | > n -1 2 -η j }. A change of variables gives I n 1 ,...,n k = σ -k D -1/2 n 1 ,...,n k R k e -|r|
n i , N (j)
n j ) i,j≤k , and denote by µ n 1 ,...,n k its smallest eigenvalue.

Let now θ ∈ (0, 1), 0 < θ 0 < θ 4 and γ > 0 be such that 2η + θ 0 + 3γ(k -1) < 1/4. We know that on Ω n 1 ,...,n k ,

tr(A n 1 ,...,n k ) = k i=1 y N (i) n i (y) 2 ≤ kn 3/2+3γ (1 + o k (1)).
We deduce that all eigenvalues of A n 1 ,...,n k are smaller than the right hand side of the above inequality. In particular on Ω n 1 ,...,n k (n -θ 0 ), there exists a constant c > 0 (depending only on k and the t i 's), such that

µ n 1 ,...,n k ≥ D n 1 ,...,n k (k n 3 2 +3γ (1 + o k (1))) k-1 ≥ c n 3 2 -θ 0 -3γ(k-1) (1 -o k (1)).
Then we get

µ n 1 ,...,n k n -1-2η ≥ n 1 2 -2η-θ 0 -3γ(k-1) (1 -o k (1)) ≥ n 1/4 (1 -o k (1)), since 2η + θ 0 + 3γ(k -1) < 1/4 by hypothesis. Note moreover, that Q n 1 ,...,n k (θ 1 , . . . , θ k ) ≥ µ n 1 ,...,n k |(θ 1 , . . . , θ k )| 2 2 . Therefore a change of variables gives J n 1 ,...,n k ≤ µ -k/2 n 1 ,...,n k {|r| 2 ≥µ 1/2 n 1 ,...,n k n -1/2-η } e -σ 2 |r| 2 2 /2 dr,
and it then follows from the first part of Lemma 20 that E

[J n 1 ,...,n k ] = o k (n -3k/4
). This concludes the proof of the lemma.

5.5. Proof of Proposition 17. Let θ 0 ∈ (0, 1/4) be fixed. Consider the events

H(θ 1 , . . . , θ k ) := {|θ 1 N (1) n 1 (y) + • • • + θ k N (k)
n k (y)| ≤ ε 0 for all y ∈ Z}, where ε 0 is as in (30), and

H(θ 1 , . . . , θ k ) := H(θ 1 , . . . , θ k ) ∩ Ω n 1 ,...,n k (n -θ 0 ).
We prove now the result for some general k ≥ 1. Fix some θ ∈ (0, 1/4), and write

n -k 4 E N n (0) k = n -k 4 n 1 ,...,n k ≤n P (Z n 1 = • • • = Z n k = 0) = n -k 4 n 1 ,...,n k ≤⌊n/d 0 ⌋ P (Z n 1 d 0 = • • • = Z n k d 0 = 0) = k! n 3k 4 0≤u 1 ≤•••≤u k ≤ 1 d 0 P Z ⌊nu 1 ⌋d 0 = • • • = Z ⌊nu k ⌋d 0 = 0 du 1 . . . du k = k! n 3k 4 n θ-1 ≤u 1 ≤•••≤u k ≤ 1 d 0 P Z ⌊nu 1 ⌋d 0 = • • • = Z ⌊nu k ⌋d 0 = 0 du 1 . . . du k + o(1).
Indeed, for the last equality, we use Theorem 5 which implies that for any ℓ ≥ 1,

n 3k 4 0≤u 1 ≤•••≤u ℓ ≤n θ-1 ≤u ℓ+1 ≤•••≤u k ≤ 1 d 0 P Z ⌊nu 1 ⌋d 0 = • • • = Z ⌊nu k ⌋d 0 = 0 du 1 . . . du k ≤ C n 3ℓ 4 +(θ-1)ℓ 0≤u ℓ+1 ≤•••≤u k ≤ 1 d 0 (u ℓ+1 . . . u k ) -3 4 = o(1),
since θ < 1/4 and where C is the constant appearing in Theorem 5. Then, by using again Theorem 5, we can apply the Lebesgue dominated convergence theorem, and we get

n -k 4 n 1 ,...,n k P (Z n 1 = • • • = Z n k = 0) = k! d σ k 0≤u 1 ≤u 2 ≤•••≤u k ≤1/d 0 p k,u 1 d 0 ,u 2 d 0 ,...,u k d 0 (0, . . . , 0) du + o(1) = d σ k [0,1/d 0 ] k p k,u 1 d 0 ,u 2 d 0 ,...,u k d 0 (0, . . . , 0) du + o(1) = d d 0 σ k [0,1] k p k,u 1 ,u 2 ,...,u k (0, . . . , 0) du + o(1) = d d 0 σ k M k,1 (0) + o(1).
This concludes the proof of the corollary.

We notice that similar calculations show that for any r ≥ 1, any k 1 , . . . , k r ≥ 1, and any

0 < t 1 < • • • < t r , E N [nt 1 ] (0) k 1 . . . N [ntr] (0) kr ∼ d σd 0 k 1 +•••+kr n k 1 +•••+kr 4 E L t 1 (0) k 1 . . . L tr (0) kr , (31)
as n → ∞.

At this point, it is also not difficult to see that the sequence (N [nt] (0)/n 1/4 , t ≥ 0) is tight in the Skorokhod space D(R). For this, notice that for any T > 0 and p ≥ 1, there exists a constant C = C(T, p), such that for all t ∈ [0, T ], h > 0, and η > 0,

P N [n(t+h)] (0) -N [nt] (0) ≥ η n 1/4 ≤ η -p n -p/4 E N [n(t+h)] (0) -N [nt] (0) p ≤ C η -p h p/4 .
Indeed the second inequality follows from the proof of Corollary 6. Since N [nt] (0) is a nondecreasing process, the tightness follows for instance from Lemma (1.7) p.517 in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF].

Proof of Proposition 7

It was proved by Kesten and Spitzer [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] that the normalized self-intersection local time of the random walk converges in distribution to its continuous counterpart. A similar convergence is proved for the mutual intersection local time in Chen's book [START_REF] Chen | Random Walk Intersections: Large Deviations and Related Topics[END_REF]. We prove Proposition 7 by following carefully their proof. [n j ′ •] / √ n before time 1 is defined by: T (j,j ′ ) n j ,n j ′ := √ n n j n j ′ N (j) n j , N (j ′ )

n j ′ = √ n n j n j ′ x∈Z n j k=1 n j ′ ℓ=1 1 {S (j) k =x} 1 {S (j ′ ) ℓ =x} .
For any ε > 0, consider the regularizing functions p ε (x) := e -x 2 /2ε / √ 2πε, and set

T (j,j ′ ) ε,n j ,n j ′ := 1 √ nn j n j ′ x∈Z n j k=1 n j ′ ℓ=1 p ε S (j) k -x √ n p ε S (j ′ ) ℓ -x √ n .
Similarly, let n j /n→T j , n j ′ /n→T j ′ E |T (j,j ′ ) n j ,n j ′ -T (j,j ′ ) ε,n j ,n j ′ | 2 = 0.

Lemma 24. (Theorem 2.2.3 in Chen) For all j = j ′ , The sequence (Λ ε,j,j ′ , ε > 0) converges in L 2 to Λ j,j ′ , as ε goes to 0.

We can then already deduce the following:

Lemma 25. For any m 1 , . . . , m k ≥ 1 and any -∞ < a j,ℓ < b j,ℓ < ∞ (j = 1, . . . , k and ℓ = 1, . . . , m j ), T (j) n j (a j,ℓ , b j,ℓ ) j=1,...,k, ℓ=1,...,m j , T (j,j ′ ) n j ,n j ′ 1≤j<j ′ ≤k , converges in distribution to (Λ j (a j,ℓ , b j,ℓ )) j=1,...,k, ℓ=1,...,m j , Λ j,j ′ 1≤j<j ′ ≤k , as n → +∞, and n j /n → T j for all j ≤ k.

Proof of Lemma 25. Let θ j,ℓ (for j = 1, . . . , k and ℓ = 1, . . . , m j ) and θ j,j ′ (for 1 ≤ j < j ′ ≤ k) be some fixed real numbers. It suffices to prove that Lemmas 23 and 24 show that we can replace the T (j,j ′ ) n j ,n j ′ and Λ j,j ′ , respectively by T (j,j ′ ) ε,n j ,n j ′ and Λ ε,j,j ′ .

Observe now that the map Therefore the weak convergence of S (j)

[n j •] / √ n, j ≤ k toward (B (j)

T j • , j ≤ k), implies that k j=1 m j ℓ=1 θ j,ℓ T (j) n j (a j,ℓ , b j,ℓ ) + 1≤j<j ′ ≤k θ j,j ′ T (j,j ′ ) ε,n j ,n j ′ , converges in distribution to k j=1 m j ℓ=1 θ j,ℓ Λ j (a j,ℓ , b j,ℓ ) + 1≤j<j ′ ≤k θ j,j ′ Λ ε,j,j ′ .

The result follows.

We finish now the proof of Proposition 7. Let θ j (for j = 1, . . . , k) and θ j,j ′ (for 1 ≤ j < j ′ ≤ k) be some fixed real numbers. We proceed like in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] by decomposing the set of all possible indices into small slices where sharp estimates can be made. Define, in the slice [τ ℓ √ n, τ (ℓ + 1) √ n), an average occupation time by T j (τ, ℓ, n) := n j n T (j) n j (τ ℓ, τ (ℓ + 1)). N (j) n j (x) 2 + 1≤j<j ′ ≤k θ j,j ′ n j n j ′ n 2 T (j,j ′ ) n j ,n j ′ -U (τ, M, n) -V (τ, M, n)

= k j=1 θ j n -3 2
-M ≤ℓ<M a(ℓ,n)≤x<a(ℓ+1,n)

N (j) n j (x) 2 - n 2 × (T j (τ, ℓ, n)) 2 (τ √ n) 2 + o(1).
It follows from computations in [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] (see in particular Lemmas 1, 2 and 3) that A(τ, M, n) converges in probability to zero as M τ 3/2 → 0. Moreover, P(U (τ, M, n) = 0) ≤ P ∃j ≤ k : sup

m≤n j |S (j) m | > M τ √ n ,
and it is well known that the right hand term goes to 0, as M τ → ∞, and n j /n → T j , for all j ≤ k.

1. 2 . 1 .

 21 Local time of Brownian motion in random scenery. Let T 1 , . . . , T k , be k positive reals. SetD T 1 ,...,T k := det(M T 1 ,...,T k ) with M T 1 ,...,T k = L T i , L T j 1≤i,j≤k , where •, • denotes the usual scalar product on L 2 (R), and C T 1 ,...,T k := E D -1/2

σ 2 4

 4 Qn 1 ,...,n k (θ 1 ,...,θ k ) Q n 1 ,...,n k (θ 1 , . . . , θ k ) = o(1) × e -σ 28 Qn 1 ,...,n k (θ 1 ,...,θ k ) .

|r| 2 2 / 8

 28 dr, at least when D n 1 ,...,n k > 0. The result now follows from Lemmas 15 and 20.

For j = 1 ,

 1 ..., k, and a < b, letT (j)n j (a, b)⌊n j •⌋ / √ n in [a, b) before time n j . The mutual intersection local time of S(j)[n j •] / √ n and S (j ′ )

T

  j (x) dx, denotes the time spent by B(j) in [a, b) before time T j , and letΛ j,j ′ := 1 T j T j ′ R L (j) T j (x)L (j ′ ) T j ′ (x) dx,denotes the mutual intersection local time of B (j)T j • and B (j ′ ) T j ′ • . Finally set for every ε > 0, Λ ε,j,j ′ := R [0,1] 2 p ε (B sT jx)p ε (B tT j ′x) ds dt dx.We will use the following lemmas:Lemma 23. (Lemma 5.3.1 in Chen ) For all j = j ′ ,

  ℓ T (j) n j (a j,ℓ , b j,ℓ ) + i 1≤j<j ′ ≤k θ j,j ′ T (j,j ′ ) n j ,n j ′ ℓ Λ j (a j,ℓ , b j,ℓ ) + i 1≤j<j ′ ≤k θ j,j ′ Λ j,j ′

  j,ℓ ≤x (j) s <b j,ℓ ] ds+ 1≤j<j ′ ≤k θ j,j ′ R [0,1] 2 p ε (x (j) sx)p ε (x (j ′ ) t x) ds dt dx,is continuous on D([0, 1], R k ) for the Skorokhod topology. Observe moreover, that for all fixed ε > 0, T (j,j ′ ) ε,n j ,n j ′ = j ′ t] √ n x ds dt dx + o(1).

  (T j (τ, ℓ, n)) 2 + 1≤j<j ′ ≤k θ j,j ′ n j n j ′ n 2 T (j,j ′ ) n j ,n j ′ ,

Now,T

  Lemma 25 shows that V (τ, M, n) converges in law to j ′ (x) dx.But the map x → L (j) t (x) being a.s. continuous with compact support, this last sum converges, as τ → 0 and M τ → ∞, to )T j ′ (x) dx.The proposition follows.

  Since D t 1 ,...,t k = D T 1 ,...,T k , Theorem 1 is equivalent to proving the existence of constants c > 0 and C > 0, such thatc (t 1 . . . t k ) -3/4 ≤ C t 1 ,...,t k ≤ C (t 1 . . . t k ) -3/4 ,(7)for all positive t 1 , . . . , t k .Let us first notice that D t 1 ,...,t k is a Gram determinant and is thus nonnegative. So C t 1 ,...,t k is well defined as an extended real number.

		j≤k	,
	and	C t 1 ,...,t k := E D t 1 ,...,t k . -1/2

  Now Proposition 8 shows that all moments of 1/λ t 1 ,...,t 2p are bounded by positive constants, uniformly in (t 1 , . . . , t 2p ). Therefore by using that for all η, M t 1 ,...,t 2p η, η ≥ λ t 1 ,...,t 2p |η| 2 , and the change of variables η → η/(λ t 1 ,...,t 2p ) 1/2 , we get for all γ < 1/6,

1

2 M t 1 ,...,t 2p η,η |η| 2γp , for some constant c p,T > 0.

  sup n θ ≤n 1 ,...,n k ≤n P Ω c n 1 ,...,n k (ε) ≤ sup n θ ≤n 1 ,...,n k ≤n P(F c n,n 1 ,...,n k ) + P det M t 1 ,...,t k ≤ 2ε ≤ O(n -θ 0 L ) + P λ t 1 ,...,t k ≤ (2ε)

  2 2 /2 dr.=⇒ D T 1 ,...,T k ,as n → ∞ and n i /n → t i , for i = 1, ..., k. This, combined with Lemma 20, shows thatE[I n 1 ,...,n k ] = √ 2π σ k C t 1 ,...,t k n -3k/4 + o k (n -3k/4 ),and it just remains to estimate E[J n 1 ,...,n k ].First consider the matrix A n 1 ,...,n k := ( N

	According to Proposition 7, we know that
	n -3k/4 D n 1 ,...,n k	(L)
	(i)

ANR project MEMEMO2.

Then by using (30) and the argument at the end of the proof of Lemma 22 we get

and, thanks to Lemma 20,

On the other hand by using the Hölder continuity of the local time (see Lemma 15), we get

Finally by using again (30), we obtain

The proposition now follows with Lemma 20.

Proof of Corollary 6

We first observe that for k = 1, the result follows from (3), since we can write