Limit theorems for additive functionals of random walks in random scenery

TL;DR

This paper investigates the asymptotic behavior of additive functionals of random walks in random scenery, establishing moment bounds and convergence results under different conditions.

Contribution

It provides new bounds for local time moments and proves distributional convergence of additive observables with specific normalizations.

Findings

Convergence in distribution of additive functionals with normalization n^(1/4).

Convergence in moments when the sum of the observable is null, with normalization n^(1/8).

Bounds for moments of the local time of the Kesten-Spitzer process.

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 n^(1/4)).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 n^(1/8)).