A functional approach for random walks in random sceneries

TL;DR

This paper introduces a functional approach to analyze random walks in random sceneries, providing general limit theorems and a framework that unifies and extends existing results, including multiple walkers in the same scenery.

Contribution

It develops a robust, general method for studying the convergence of RWRS, allowing separate analysis of the walk's local time and the scenery's measures, and recovers known and new results.

Findings

Proved functional limit theorems for RWRS under broad conditions

Established criteria for convergence of local times of the walk

Analyzed convergence of scenery-related random measures

Abstract

A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study separately the convergence of the walk and of the scenery: on the one hand, a general criterion for the convergence of the local time of the walk is provided, on the other hand, the convergence of the random measures associated with the scenery is studied. This functional approach is robust enough to recover many of the known results on RWRS as well as new ones, including the case of many walkers evolving in the same scenery.