Moment convergence of first-passage times in renewal theory

Let ξ1,ξ2,… be independent copies of a positive random variable ξ, S0=0, and Sk=ξ1+…+ξk, k∈N. Define N(t)=inf{k∈N:Sk>t} for t≥0. The process (N(t))t≥0 is the first-passage time process associated with (Sk)k≥0. It is known that if the law of ξ belongs to the domain of attraction of a stable law or P(ξ>t) varies slowly at ∞, then N(t), suitably shifted and scaled, converges in distribution as t→∞ 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.