Power and exponential moments of the number of visits and related quantities for perturbed random walks

TL;DR

This paper investigates the conditions under which certain random quantities related to perturbed random walks, such as exit times and visit counts, are finite or have finite exponential or power moments, providing theoretical criteria.

Contribution

It offers new criteria for the almost sure finiteness and moment finiteness of key quantities in perturbed random walks, extending understanding of their probabilistic behavior.

Findings

Criteria for almost sure finiteness of exit times and visit counts.

Conditions for finiteness of exponential moments of these quantities.

Criteria for the finiteness of power moments of visit-related quantities.

Abstract

Let $(\xi_1,\eta_1),(\xi_2,\eta_2),...$ be a sequence of i.i.d.\ copies of a random vector $(\xi,\eta)$ taking values in $\R^2$, and let $S_n := \xi_1+...+\xi_n$. The sequence $(S_{n-1} + \eta_n)_{n \geq 1}$ is then called perturbed random walk. We study random quantities defined in terms of the perturbed random walk: $\tau(x)$, the first time the perturbed random walk exits the interval $(-\infty,x]$, $N(x)$, the number of visits to the interval $(-\infty,x]$, and $\rho(x)$, the last time the perturbed random walk visits the interval $(-\infty,x]$. We provide criteria for the a.s.\ finiteness and for the finiteness of exponential moments of these quantities. Further, we provide criteria for the finiteness of power moments of $N(x)$ and $\rho(x)$.