Limit theorems for additive functionals of random walks in random scenery

Françoise Pène

doi:10.1214/21-ejp696

Abstract

We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process. These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n1 4). When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n1 8).

Highlights

1.1 Description of the model and of some earlier resultsWe consider two independent sequences (Xk)k≥1 andy∈Z of independent identically distributed Z-valued random variables
Limit theorems for additive functionals of random walks in random scenery where we set Nn(y) = #{k = 0, . . . , n − 1 : Sk = y} for the local time of S at position y before time n
This process first studied by Borodin [7] and Kesten and Spitzer

Summary

Description of the model and of some earlier results

We consider two independent sequences (Xk)k≥1 (the increments of the random walk) and (ξy)y∈Z (the random scenery) of independent identically distributed Z-valued random variables. N − 1 : Sk = y} for the local time of S at position y before time n This process first studied by Borodin [7] and Kesten and Spitzer [32] describes the evolution of the total amount won until time n by a particle moving with respect to the random walk S, starting with a null amount at time 0 and winning the amount ξ at each time the particle visits the position ∈ Z. This process is a natural example of (strongly) stationary process with long time dependence.

Main results

Upper bound for moments

Law of large numbers

Proof of the central limit theorem: proof of Theorem 5

B Moment convergence in Theorem 3