Tail asymptotics for the supremum of a random walk when the mean is not finite

Abstract

We consider the sums $S_n={\xi }_1+\cdots +{\xi }_n$ of independent identically distributed random variables. We do not assume that the $\xi$'s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability $\mathbf{P}\{M>x\}$ as $x\to \infty$, provided that $M=\sup \{S_n,n\ge 1\}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that subexponentiality of distribution $F$ does not imply subexponentiality of its integrated tail distribution $F^I$.