Renewal theorems for random walks in random scenery

TL;DR

This paper establishes renewal theorems for random walks in random scenery, analyzing their asymptotic behavior when the underlying distributions are in the domain of attraction of stable laws.

Contribution

It provides new renewal theorems for random walks in random scenery with stable domain of attraction assumptions, extending previous results to broader distribution classes.

Findings

Asymptotic behavior characterized for sums involving $h(Z_n - a)$ as $|a|$ grows.

Results apply to both recurrent and transient regimes of the process.

Extension of renewal theory to processes with stable law distributions.

Abstract

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 $\mathbb{Z}$.