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 = ΣN 1 Xi 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.

Branching processes, Convolution, Regular variation, Subexponential distributions,
Stochastic Processes and Their Applications
Erasmus School of Economics

Geluk, J.L. (1996). Tails of subordinated laws: The regularly varying case. Stochastic Processes and Their Applications, 61(1), 147–161. doi:10.1016/0304-4149(95)00070-4