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

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

2011-11-17

arxiv.org/abs/1111.4159v2

Mathematics

Probability

36 pages

Scientific paper

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 $\rho(x)$.

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