Exponential moments of first passage times and related quantities for random walks

Abstract

For a zero-delayed random walk on the real line, let $\tau (x)$, $N(x)$ and $\rho (x)$ denote the first passage time into the interval $(x,\infty )$, the number of visits to the interval $(-\infty ,x]$ and the last exit time from $(-\infty ,x]$, respectively. In the present paper, we provide ultimate criteria for the finiteness of exponential moments of these quantities. Moreover, whenever these moments are finite, we derive their asymptotic behaviour, as $x\to \infty$.