Asymptotics of randomly stopped sums in the presence of heavy tails

Abstract

We study conditions under which $$\mathbf{P}\{S_τ >x\} ∼ \mathbf{P}\{M_τ > x\} ∼ \mathbf{E}τ\mathbf{P}\{ξ_1 > x\} \mbox{as } x → ∞,$$ where $S_τ$ is a sum $ξ_1 + ⋯ + ξ_τ$ of random size $τ$ and $M_τ$ is a maximum of partial sums $M_τ = \max_{n≤τ} S_n$. Here, $ξ_n, n = 1, 2, …,$ 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 $\mathbf{E}ξ > 0$ and where the tail of $τ$ is comparable with, or heavier than, that of $ξ$, and obtain the asymptotics $$\mathbf{P}\{S_τ > x\} ∼ \mathbf{E}τ\mathbf{P}\{ξ_1 > x\} + \mathbf{P}\{τ > x / \mathbf{E}ξ\} \mbox{as } x → ∞.$$ 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 $\mathbf{P}\{S_n > x\} / \mathbf{P}\{ξ_1 > x\}$ which substantially improve Kesten’s bound in the subclass ${\mathcal{S}}^{*}$ of subexponential distributions.