Compound Poisson approximation for simple transient random walks in random sceneries

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.