Second order behaviour of the tail of a subordinated probability distribution

AbstractLet G = Σ∞n=0pnF∗n denote the probability measure subordinate to F with subordinator {Pn}N. We investigate the asymptotic behaviour of (1 − G(x))−(Σ npn)(1 − F(x)) as x → ∞ if 1 − F is regularly varying with index ϱ, 0 ≤ ϱ ≤ 1. Applications to random walk theory and infinite divisibility are given