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

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 $\rho(x)$.