Asymptotics of randomly stopped sums in the presence of heavy tails

Denisov, Denis and Foss, Sergey and Korshunov, Dmitry (2010) Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli, 16 (4). pp. 971-994. ISSN 1350-7265

Abstract

We study conditions under which P{Sτ > x} ∼ P{Mτ > x} ∼ EτP{ξ1 > x}  as x → ∞, where Sτ is a sum ξ1 + ⋯ + ξτ of random size τ and Mτ is a maximum of partial sums Mτ = maxn≤τ Sn. Here, ξn, n = 1, 2, …, are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where τ is independent of the summands; also, in a particular situation, we deal with a stopping time. We also consider the case where Eξ > 0 and where the tail of τ is comparable with, or heavier than, that of ξ, and obtain the asymptotics P{Sτ > x} ∼ EτP{ξ1 > x} + P{τ > x / Eξ}  as x → ∞. This case is of primary interest in branching processes. In addition, we obtain new uniform (in all x and n) upper bounds for the ratio P{Sn > x} / P{ξ1 > x} which substantially improve Kesten’s bound in the subclass ${\mathcal{S}}^{*}$ of subexponential distributions.