Empirical processes for recurrent and transient random walks in random scenery

TL;DR

This paper studies the asymptotic behavior of empirical processes generated by random walks in random scenery, extending previous work to include transient walks and recurrent walks with specific stable limit distributions.

Contribution

It generalizes prior results by analyzing empirical processes for both transient and recurrent random walks in random scenery with new limit theorems.

Findings

Established limit processes for transient random walks in ^d.

Derived asymptotic distributions for recurrent walks with stable limits.

Extended empirical process theory to new classes of random walks.

Abstract

In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where $(\xi_x, x\in\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\in\mathbb N}$ is a random walk evolving in $\mathbb{Z}^d$, independent of the $\xi$'s. In Wendler (2016), the case where $(S_n)_{n\in\mathbb N}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac 1\alpha}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $\alpha$, with $\alpha\in(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{n\in\mathbb N}$ is either: a) a transient random walk in $\mathbb{Z}^d$, b) a recurrent random walk in $\mathbb{Z}^d$ such that $(n^{-\frac 1d}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $d\in\{1,2\}$.