Point processes of exceedances for random walks in random sceneries

TL;DR

This paper studies the limit behavior of exceedance point processes for a sequence derived from a random walk in a random scenery, showing convergence to a compound Poisson process and analyzing extremal properties.

Contribution

It extends previous maximum limit theorems to point process convergence and explores extremal indices for the sequence in a random walk setting.

Findings

Exceedance point processes converge to a compound Poisson process.

The extremal index for the sequence is characterized.

Weak mixing properties are discussed.

Abstract

Let $\{\xi(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables and let $\{S_n, n \in \mathbb{N}_+ \}$ be a transient random walk in the domain of attraction of a stable law. In the previous work \cite{Nicolas_Ahmad}, under conditions of type $D(u_n)$ and $D'(u_n)$ we provided a limit theorem for the maximum of the first $n$ terms of the sequence $\{\xi(S_n), n \in \mathbb{N} \}$. In this paper, under the same conditions we will see that, the limit of the process which counts the numbers of the exceedances of the form $\{\xi(S_k)>u_n\}, k\geq 1$, is a compound Poisson point process. We also deal with the so-called extremal index for the sequence $\{\xi(S_n), n \in \mathbb{N} \}$ and we discuss some weak mixing properties.