Some properties on extremes for transient random walks in random sceneries

TL;DR

This paper studies the extreme values of a stationary sequence observed along a transient random walk, showing convergence of exceedance point processes to a Poisson process and exploring properties of the sequence.

Contribution

It extends previous limit theorems by demonstrating the convergence of exceedance point processes to a Poisson process under certain conditions.

Findings

Exceedance point process converges to a Poisson process.

Properties of the sequence $(\xi(S_n))_{n ext{≥}0}$ are established.

Results hold under conditions $D(u_n)$ and $D'(u_n)$.

Abstract

Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction of a stable law and let $(\xi(s))_{s \in \mathbb{Z}}$ be a stationary sequence of random variables. In a previous work, under conditions of type $D(u_n)$ and $D'(u_n)$, we established a limit theorem for the maximum of the first $n$ terms of the sequence $(\xi(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point process of exceedances converges to a Poisson point process. We also give some properties of $(\xi(S_n))_{n\geq 0}$.