Moment convergence of first-passage times in renewal theory

Let ${\xi }_1,{\xi }_2,\dots$ be independent copies of a positive random variable $\xi$, $S_0=0$, and $S_k={\xi }_1+\dots +{\xi }_k$, $k\in \mathbb{N}$. Define $N(t)=\inf \{k\in \mathbb{N}:S_k>t\}$ for $t\ge 0$. The process $(N(t){)}_{t\ge 0}$ is the first-passage time process associated with $(S_k{)}_{k\ge 0}$. It is known that if the law of $\xi$ belongs to the domain of attraction of a stable law or $\mathbb{P}(\xi >t)$ varies slowly at $\mathrm{\infty }$, then $N(t)$, suitably shifted and scaled, converges in distribution as $t \to \infty$toarandomvariable$W$ with a stable law or a Mittag-Leffler law. We investigate whether there is convergence of the power and exponential moments to the corresponding moments of $W$. Further, the analogous problem for first-passage times of subordinators is considered.