Asymptotics of randomly stopped sums in the presence of heavy tails

We study conditions under which P{S-tau > x} similar to P{M-tau > x} similar to E tau P{xi(1) > x} as x -> infinity, 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 <=tau) S-n. Here, xi(n), n= 1, 2, ... , are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where tau is independent of the summands; also, in a particular situation, we deal with a stopping time. We also 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) similar to E tau P{xi(1) > x} + P{tau > x/E xi} as x -> infinity. 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{S-n > x}/P{xi(1) > x} which substantially improve Kesten's bound in the subclass S* of subexponential distributions.