Extremes for transient random walks in random sceneries under weak independence conditions

TL;DR

This paper establishes a limit theorem for the maximum of a sequence sampled along a transient random walk in a weakly dependent stationary sequence, extending previous results from the i.i.d. case.

Contribution

It generalizes the extreme value theory for maxima of sequences sampled along random walks to weakly dependent stationary sequences.

Findings

Derived a limit theorem for the maximum of the sequence along the random walk.

Extended previous i.i.d. results to weak dependence conditions.

Applicable to sequences satisfying $D(u_n)$ and $D'(u_n)$ conditions.

Abstract

Let $\{\xi(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables with conditions of type $D(u_n)$ and $D'(u_n)$. Let $\{S_n, n \in \mathbb{N} \}$ be a transient random walk in the domain of attraction of a stable law. We provide a limit theorem for the maximum of the first $n$ terms of the sequence $\{\xi(S_n), n \in \mathbb{N} \}$ as $n$ goes to infinity. This paper extends a result due to Franke and Saigo who dealt with the case where the sequence $\{\xi(k), k \in \mathbb{Z} \}$ is i.i.d.