Moment convergence of first-passage times in renewal theory

Let xi(1),xi(2),... be independent copies of a positive random variable xi, S-0 = 0, and S-k = set xi(1) + ... + xi(k), k is an element of N. Define N(t) = inf{k is an element of N : S-k > t} for t >= 0. The process (N(t))(t >= 0) is the first-passage time process associated with (S-k)(k >= 0). It is known that if the law of xi belongs to the domain of attraction of a stable law or P( xi > t) varies slowly at infinity, then N(t), suitably shifted and scaled, converges in distribution as t -> infinity 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. (C) 2016 Elsevier B.V. All rights reserved.