Power and Exponential Moments of the Number of Visits and Related Quantities for Perturbed Random Walks

Let $$(\xi _1,\eta _1),(\xi _2,\eta _2),\ldots $$ be a sequence of i.i.d. copies of a random vector $$(\xi ,\eta )$$ taking values in $$\mathbb{R }^2$$ , and let $$S_n:= \xi _1+\cdots +\xi _n$$ . 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 almost sure 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 $$\rho (x)$$ . In the course of the proofs of our main results, we investigate the finiteness of power and exponential moments of shot-noise processes and provide complete criteria for both, power and exponential moments.