Compound Poisson approximation for simple transient random walks in random sceneries

Abstract

Given a simple transient random walk $(S_n{)}_{n\ge 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(\xi (s){)}_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(\xi (S_n){)}_{n\ge 0}$. 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.