Asymptotics of randomly stopped sums in the presence of heavy tails

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 ${\mathcal{S}}^{\ast }$ of subexponential distributions.

22 pages

References in corpus (1)

• Lower limits and equivalences for convolution tails