Moment convergence of first-passage times in renewal theory

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