Tails of subordinated laws: The regularly varying case

Suppose Xi, i = 1, 2, … are i.i.d. positive random variables with d.f. F. We assume the tail d.f. F̄ = 1 − F to be regularly varying (F̄(tx)F̄(t) → x−β, x > 0, t → ∞) with 0 < β < 1. The asymptotic behaviour of P(SN > x) as x → ∞ where SN = ΣN1Xi and N,Xi(i ≥ 1) independent with Σ∞n=0P(N = n)xn analytic at x = 1 is studied under an additional smoothness condition on F. As an application we give the asymptotic behaviour of the expected population size of an age-dependent branching process.