Renewal theorems for random walks in random scenery

Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We suppose that the distributions of $X_1$ and $\xi_0$ belong to the normal domain of attraction of strictly stable distributions with index $\alpha\in[1,2]$ and $\beta\in(0,2)$ respectively. We are interested in the asymptotic behaviour as $|a|$ goes to infinity of quantities of the form $\sum_{n\ge 1}{\mathbb E}[h(Z_n-a)]$ (when $(Z_n)_n$ is transient) or $\sum_{n\ge 1}{\mathbb E}[h(Z_n)-h(Z_n-a)]$ (when $(Z_n)_n$ is recurrent) where $h$ is some complex-valued function defined on $\mathbb{R}$ or $\mathbb{Z}$.

given (see Chapter VII in [19]). In this paper we are interested in renewal theorems for random walk in random scenery (RWRS). Random walk in random scenery is a simple model of process in disordered media with long-range correlations. They have been used in a wide variety of models in physics to study anomalous dispersion in layered random flows [17], diffusion with random sources, or spin depolarization in random fields (we refer the reader to Le Doussal's review paper [15] for a discussion of these models). On the mathematical side, motivated by the construction of new self-similar processes with stationary increments, Kesten and Spitzer [14] and Borodin [4,5] introduced RWRS in dimension one and proved functional limit theorems. This study has been completed in many works, in particular in [3] and [9]. These processes are defined as follows. We consider two independent sequences (X k , k ≥ 1) and (ξ y , y ∈ Z) of independent identically distributed random variables with values in Z and R respectively. We define ∀n ≥ 1, S n := n k=1 X k and S 0 := 0.
The random walk in random scenery Z is then defined for all n ≥ 1 by The symbol # stands for the cardinality of a finite set. Denoting by N n (y) the local time of the random walk S : N n (y) = # k = 1, ..., n : S k = y the random variable Z n can be rewritten as The distribution of ξ 0 is assumed to belong to the normal domain of attraction of a strictly stable distribution S β of index β ∈ (0, 2], with characteristic function φ given by φ(u) = e −|u| β (A 1 +iA 2 sgn(u)) , u ∈ R, where 0 < A 1 < ∞ and |A −1 1 A 2 | ≤ | tan(πβ/2)|. When β = 1, A 2 is null. We will denote by ϕ ξ the characteristic function of the random variables ξ x . When β > 1, this implies that E[ξ 0 ] = 0. Under these conditions, we have, for β ∈ (0, 2], Concerning the random walk (S n ) n≥1 , the distribution of X 1 is assumed to belong to the normal domain of attraction of a strictly stable distribution S ′ α of index α. Since, when α < 1, the behaviour of (Z n ) n is very similar to the behaviour of the sum of the ξ k 's, k = 1, . . . , n, we restrict ourselves to the study of the case when α ∈ [1,2]. Under the previous assumptions, the following weak convergences hold in the space of càdlàg real-valued functions defined on [0, ∞) and on R respectively, endowed with the Skorohod topology : where U and Y are two independent Lévy processes such that U (0) = 0, Y (0) = 0, U (1) has distribution S ′ α and Y (1) has distribution S β . For α ∈ ]1, 2], we will denote by (L t (x)) x∈R,t≥0 a continuous version with compact support of the local time of the process (U (t)) t≥0 and by |L| β the random variable R L β 1 (x) dx 1/β . Next let us define In [14], Kesten and Spitzer proved the convergence in distribution of ((n −δ Z nt ) t≥0 ) n , when α > 1, to a process (∆ t ) t≥0 defined as by considering a process (Y (−x)) x≥0 with the same distribution as (Y (x)) x≥0 and independent of U and (Y (x)) x≥0 .
In [9], Deligiannidis and Utev considered the case when α = 1 and β = 2 and proved the convergence in distribution of ((Z nt / n log(n)) t≥0 ) n to a Brownian motion. This result is got by an adaptation of the proof of the same result by Bothausen in [3] in the case when β = 2 and for a square integrable two-dimensional random walk (S n ) n .
In [8], Castell, Guillotin-Plantard and Pène completed the study of the case α = 1 by proving the convergence of (n − 1 β (log(n)) where a 0 is such that t → e −a 0 |t| is the characteristic function of the limit of (n −1 S n ) n .
The distribution of ξ 0 belongs to the normal domain of attraction of S β with characteristic function φ given by (2).
For any a ∈ R (resp. a ∈ Z), we consider the kernel K n,a defined as follows : for any h : R → C (resp. h : Z → C) in the strongly non-lattice (resp. in the lattice) case, we write when it is well-defined. Theorem 1. The following assertions hold for every integrable function h on R with Fourier transform integrable on R in the strongly non-lattice case and for every integrable function h on Z in the lattice case.
• when α > 1 and β > 1, • when α ≥ 1 and β = 1, • when α = 1 and β = 2, assume that h is even and that the distribution of the ξ ′ x s is symmetric, then Remarks: 1-It is worth noticing that since |A 2 /A 1 | ≤ | tan(πβ/2)|, the constants C 1 and D 1 are strictly positive. 2-The limit as a goes to −∞ is not considered in Theorem 1 and Theorem 2 since it can be easily obtained from the limit as a goes to infinity. Indeed, the problem is then equivalent to study the limit as a goes to infinity with the random variables (ξ x ) x replaced by (−ξ x ) x and the function h by x → h(−x). The limits can easily be deduced from the above limit constants by changing A 2 to −A 2 .
1.2. Transient case : β ∈ (0, 1). Let H 1 denote the set of all the complex-valued Lebesgueintegrable functions h such that its Fourier transformĥ is continuously differentiable on R, with in additionĥ and (ĥ) ′ Lebesgue-integrable.
Theorem 2. Assume that α ∈ (1, 2] and that the characteristic function of the random variable ξ 0 is equal to φ given by (2). Then, for all h ∈ H 1 , we have

1.3.
Preliminaries to the proofs. In our proofs, we will use Fourier transforms for some h : R → C or h : Z → C and, more precisely, the following fact This will lead us to the study of n≥1 E[e itZn ]. Therefore it will be crucial to observe that we have since, taken (S k ) k≤n , (ξ y ) y is a sequence of iid random variables with characteristic function ϕ ξ . Let us notice that, in the particular case when ξ 0 has the stable distribution given by characteristic function (2), the quantity n≥1 E[e itZn ] is equal to Section 2 is devoted to the study of this series thanks to which, we prove Theorem 1 in Section 3 and Theorem 2 in Section 4.

Study of the series Ψ
Let us notice that we have, for every real number t = 0, with V n := y∈Z N n (y) β . Let us observe that Nn(y) Combining this with the fact that y∈Z N n (y) = n, we obtain: Proposition 3. When β ∈ (0, 2], for every r ∈ (0, +∞), the function ψ is bounded on the set {t ∈ R : |t| ≥ r}. When β ∈ (0, 1), the function ψ is differentiable on R \ {0}, and for every r ∈ (0, +∞), its derivative ψ ′ is bounded on the set {t ∈ R : |t| ≥ r}.
Next, when β ∈ (0, 1), since n≥1 n e −A 1 (rn) β < ∞, it easily follows from Lebesgue's theorem In the particular case when β = 1, we have A 2 = 0 and When β = 1, the expression of ψ(t) is not so simple. We will need some estimates to prove our results. Recall that the constant c is defined in (5) if α = 1 and set where γ is the function defined by When α = 1 and β > 1, we have where γ is the function defined by .
To prove Proposition 4, we need some preliminaries lemmas. Let us define We first recall some facts on the behaviour of the sequence Lemma 5 (Lemma 6 in [14], Lemma 5 in [8]). When α > 1, the sequence of random variables converges almost surely to c 1 β .
Lemma 6 (Lemma 11 in [7], Lemma 16 in [8]). If β > 1, then If β ≤ 1, then for every p ≥ 1, The idea will be that V n is of order b β n . Therefore the study of n≥1 e −|tbn| β (A 1 +iA 2 sgn(t)) will be useful in the study of ψ(t). For any function g : R + → R, we denote by L(g) the Laplace transform of g given, for every z ∈ C with Re(z) > 0, by when it is well defined.
Lemma 7. When α > 1, for every complex number z such that Re(z) > 0 and every p ≥ 0, we have When α = 1, for every complex number z such that Re(z) > 0 and every p ≥ 0, we have Here w is the Lambert function defined on [0; +∞) as the inverse function of y → ye y (defined on [0; +∞)) and ∆ is the function defined on [e; +∞) as the inverse function of y → e y /y defined on [1; +∞).
Proof. First, we consider the case when α > 1. With the change of variable y = (ux) δβ , we get Let us write By applying Taylor's inequality to the function v → e −vz v p on the interval [(u ⌊x⌋) δβ , (ux) δβ ], we obtain for every x > 2 (use ⌊x⌋ ≥ x/2) Next, by applying Taylor's inequality to the function t → t δβ according that δβ > 1 or δβ < 1 (again use ⌊x⌋ ≥ x/2 in the last case), we have Therefore, with the change of variable t = (ux/2) δβ , we get Now, we suppose that α = 1 and follow the same scheme. We observe that δβ = 1. With the change of variable t = x(log x) β−1 , we get y(1+w(y)) and since e w(x) = x w(x) ). Moreover, if β < 1, we have y(∆(y)−1) and since e ∆(y) = y∆(y)).
Applying Taylor's inequality, we get Hence there exists some c depending only on p and |z| such that The last integral is finite.
Lemma 8. For every complex number z such that Re(z) > 0 and every p ≥ 0, we have Proof. We know that w(x) ∼ +∞ log x and ∆(x) ∼ +∞ log x. Hence, for every p ≥ 0, we havẽ Now, we apply Tauberian theorems (Theorems p. 443-446 in [12]) to Laplace transforms L(.) defined for complex numbers such that Re(z) > 0. The lemma follows.
There exist a sequence of random variables (a n ) n and a random variable A defined on (0, 1) endowed with the Lebesgue measure λ such that, for every n ≥ 1, a n and have the same distribution, such that E λ [sup n≥1 a n ] < +∞ and such that (a n ) n converges almost surely and in L 1 to the random variable A.
Proof. Lemmas 5 and 6 insure in particular the uniform integrability of (b n V − 1 β n ) 1 δ n and the existence of a sequence of random variables (a n ) n and of a random variable A defined on (0, 1) endowed with the Lebesgue measure λ such that, for every n ≥ 1, a n and (b n V − 1 β n ) 1 δ have the same distribution, such that A has the same distribution as |L| −1/δ β if α > 1 and is equal to c −1 if α = 1, such that E λ [sup n≥1 a n ] < +∞ and (a n ) n converges almost surely to A and so in L 1 . Indeed, following the the Skorohod representation theorem, we define a n (x) := inf u > 0 : and A as follows : A The sequence (a n ) n converges almost surely to the random variable A as n goes to infinity. Moreover, from the formula E λ [sup n≥1 a n ] = +∞ 0 λ(sup n a n > t) dt and the fact that we get λ(sup n a n > t) = sup with γ := 2 when β ≤ 1 ; γ := δβ/(β − 1) when α > 1 and β > 1 or when α = 1 and β ∈ (1, 2).
Therefore, using the previous lemma, the series ψ can be rewritten, for every real number t = 0, as Lemma 10. There exists t 0 > 0 such that when α > 1 or (α = 1, β > 1), the family of random variables   1 is uniformly integrable and such that, if α > 1, the family  is also uniformly integrable.
Proof. If α > 1, thanks to lemma 7, we know that, for every real number t ∈ (0, 1) and every complex number z such that Re(z) > 0, we have and, from which we conclude.
Lemma 11. If (α > 1, β = 1) or (α = 1, β > 1), we have Proof. We only prove the first assertion, the proof of the second one following the same scheme. Let β = 1 and α ≥ 1. Let ε ∈ (0, 1/δ), Now it remains to prove the almost sure convergence to 0 as t goes to 0 of the following quantity : By applying Taylor's inequality to the function v → e −|t| β |v| δβ b β n (A 1 +iA 2 sgn(t)) , we have using lemmas 7 and 8 and according to the fact that (a n ) n converges almost surely to A.
Proof of Proposition 4. First consider the case α > 1 and β = 1. Thanks to lemmas 7 and 11, we get that as t goes to 0. Therefore, thanks to (12) and to the uniform integrability (Lemma 10), we deduce (8). The proof of (10) is similar (using Lemma 8) and is omitted. Again, to prove (9), we use (12). Since for t = 0, as the sum of The second assertion in Lemma 11 and the uniform integrability in Lemma 10 implies that the first sum goes to 0 as t goes to 0. From Lemma 7, we get that the second one goes to 0 as t goes to 0.

Proof of Theorem 1
We first begin to prove that for every a ∈ R, the sequence of converges as n tends to infinity. Indeed, for every a ∈ R, we have is bounded on any set [r, +∞[ with r > 0 and so the series is well defined for every t = 0. ii)-We have Proof. In order to prove ii), we show that From Lemma 6 in [7] and Lemma 12 in [8], for every η > 0 and every n ≥ 1, there exists a subset Ω n such that for every p > 1, P(Ω n ) = 1 − o(n −p ) and such that, on Ω n , we have with E n (t) := y∈Z ϕ ξ (tN n (y)) − e −|t| β Vn(A 1 +iA 2 sgn(t)) .
Now, since |t| ≤ n −δ+η , on Ω n , for every y ∈ Z, we have |t|N n (y) ≤ n Now, we fix some t = 0. Let us write as t → 0, using Proposition 4 and the continuity of the function h at 0 (and the fact that . Then, ii)-is proved and i)-can easily be deduced from the above arguments.

The integrand in (15) is bounded by
Let r > 0, on the set {t; |t| ≥ r}, by i)-from Proposition 12, sinceĥ is integrable, Θ is integrable. From Propositions 4 and 12 (item ii)-) and from the fact thatĥ is continuous at 0, Θ(t) is in O (|t|γ(t)) (at t = 0), which is integrable in the neighborhood of 0 in all cases considered in Theorem 1 except (α, β) = (1, 2). From the dominated convergence theorem, we deduce that In the case (α, β) = (1, 2), by assumption, for every integer n ≥ 1, the function t →ĥ(t) n k=1 E[e itZ k ] being even, we have The integrand in (17) is uniformly bounded in n by a function in O log(1/|t|) −1 (at t = 0), which is integrable in the neighborhood of 0. From the dominated convergence theorem, we deduce that In the rest of the proof we only consider the strongly non-lattice case, the lattice case can be handled in the same way.

Let us observe that
By applying the residue theorem to the function z → z −1/δ (1 − e −iz ) with the contour in the complexe plane defined as follows : the line segment from −ir to −iR (r < R), the circular arc connecting −iR to R, the line segment from R to r and the circular arc from r to −ir and letting r → 0, R → +∞, we get that From this formula we easily deduce the first statement of theorem 1 using the fact that 2)). Moreover, by combining propositions 4 and 12, we have lim t→0 (γ(t)) −1ψ (t) − 1 = 0.
Hence, for every ε ∈ (0, 1), there exists 0 < A ε < 1 such that Sinceψ is bounded on [A ε , +∞[ andĥ is integrable on I, we have Let a be such that a ≥ A −1/β ε . We have that can be neglected as a goes to infinity since as a goes to infinity and since It remains to estimate 1 2π {a −β ≤|t|≤Aε}ĥ (t)ψ(t)(1 − e −ita )dt that we decompose into two parts: •We first estimate I 2 (a) for a large. Remark that by the change of variables u = at, We treat separately the cases β = 1 and α = 1, β ∈ (1, 2). If β = 1, we have This comes from the fact that t dt x is bounded.
The study of I 3 (a) is easy. Set g 3 (t) = (1 − χ(t))ĥ(t)ψ(t). From (21), we have I 3 (a) = g 3 (a), and from Propositions 3 and 4, g 3 and g ′ 3 are Lebesgue-integrable on R. An integration by parts then gives The next two subsections are devoted to the study of I 1 (a) and I 2 (a).
Proof. For every u = 0, we have For any x > 0, let us denote by f x the Fourier transform of the function u → e −x|u| Next, since we have: x δ , we obtain, from Fubini's theorem, with the change of variable y = |v|/x δ and finally by the dominated convergence theorem (since c ± δ,β are well defined, see below), that Let us compute c + δ,β . We have 1 δβ with the contour in the complexe plane defined as follows : the line segment from r to R (r < R), the circular arc connecting R to iR, the line segment from iR to ir and the circular arc from ir to r and letting r → 0, R → +∞, we get that Taking A → +∞, we get the expression of c + δ,β .
The desired property for I 2 (a) is then established. This completes the proof of Theorem 2.
Remark: The generalization of our proof to the more general context when the distribution of ξ 0 belongs to the normal domain of attraction of a stable distribution of index β is not as simple as in the recurrent case. Indeed we used precise estimation of the derivative of ψ that should require the existence of the derivative of ϕ ξ outside 0, which does not appear as a natural hypothesis when β < 1 since ξ 0 is not integrable.