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 $x \to \infty$.
"Exponential Moments of First Passage Times and Related Quantities for Random Walks." Electron. Commun. Probab. 15 365 - 375, 2010. https://doi.org/10.1214/ECP.v15-1569