Point processes of exceedances for random walks in random sceneries

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 $D^{\prime }(u_n)$ 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 $\{\xi (S_k)>u_n\},k\ge 1$, 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.

arXiv admin note: text overlap with arXiv:1910.04651