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

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 $x \to \infty$.