Compound Poisson approximation for simple transient random walks in random sceneries

TL;DR

This paper studies the extreme values of a sequence derived from a transient random walk in a random scenery, showing convergence to a compound Poisson process and providing explicit examples of cluster sizes.

Contribution

It explicitly characterizes the extremal index and the convergence of exceedance point processes for transient random walks in random sceneries, with detailed examples.

Findings

Extremal index is explicitly determined.

Point process of exceedances converges to a compound Poisson process.

Cluster size distributions are explicitly derived in examples.

Abstract

Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(\xi(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(\xi(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit.