Asymptotics of randomly stopped sums in the presence of heavy tails

We study conditions under which P{S_t > x} ~ P{M _t > x} ~ EtP{?₁ > x} as x?8, where S_t is a sum ?₁+ ? ?_t of random size t and Mt is a maximum of partial sums Mt = max_n=t S_n. Here, ?_n, n = 1, 2,. . ., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where t is independent of the summands; also, in a particular situation, we deal with a stopping time. We also consider the case where E? > 0 and where the tail of t is comparable with, or heavier than, that of ?, and obtain the asymptotics P{S_t > x} ~ EtP{?1 > x} + P{t > x/E?} as x?8. 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{?1 > x} which substantially improve Kesten's bound in the subclass S* of subexponential distributions. © 2010 ISI/BS.