Asymptotics of randomly stopped sums in the presence of heavy tails

TL;DR

This paper investigates the asymptotic behavior of randomly stopped sums with heavy-tailed distributions, providing new conditions and bounds that extend classical results and are particularly relevant for branching processes.

Contribution

It establishes new asymptotic equivalences for heavy-tailed sums and maxima, including cases with heavy-tailed stopping times, and improves existing bounds for subexponential distributions.

Findings

Asymptotic equivalence of tail probabilities for sums and maxima with heavy tails.

New asymptotics involving the tail of the stopping time in branching processes.

Enhanced uniform bounds for tail ratios in subexponential distributions.

Abstract

We study conditions under which $P(S_\tau>x)\sim P(M_\tau>x)\sim E\tau P(\xi_1>x)$ as $x\to\infty$, where $S_\tau$ is a sum $\xi_1+...+\xi_\tau$ of random size $\tau$ and $M_\tau$ is a maximum of partial sums $M_\tau=\max_{n\le\tau}S_n$. Here $\xi_n$, $n=1$, 2, ..., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where $\tau$ is independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where $E\xi>0$ and where the tail of $\tau$ is comparable with or heavier than that of $\xi$, and obtain the asymptotics $P(S_\tau>x) \sim E\tau P(\xi_1>x)+P(\tau>x/E\xi)$ as $x\to\infty$. This case is of a primary interest in the branching processes. In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the ratio $P(S_n>x)/P(\xi_1>x)$ which substantially improve Kesten's bound in the subclass ${\mathcal S}^*$ of subexponential distributions.