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 ${\mathbb{R}}^2$, and let $S_n:={\xi }_1+...+{\xi }_n$. The sequence $(S_{n-1}+{\eta }_n{)}_{n\ge 1}$ 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 $\rho (x)$.

arXiv comment38 pages