Moment convergence of first-passage times in renewal theory

Let $\xi_1, \xi_2, \ldots$ be independent copies of a positive random variable $\xi$, $S_0 = 0$, and $S_k = \xi_1+\ldots+\xi_k$, $k \in \mathbb{N}$. Define $N(t) = \inf\{k \in \mathbb{N}: S_k>t\}$ for $t\geq 0$. The process $(N(t))_{t\geq 0}$ is the first-passage time process associated with $(S_k)_{k\geq 0}$. 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 $\infty$, then $N(t)$, suitably shifted and scaled, converges in distribution as $t \to \infty$ to a random variable $W$ 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.