Asymptotics of randomly stopped sums in the presence of heavy tails

Abstract

We study conditions under which P{S� > x} ∼ P{M� > x} ∼ EP{�1 > x} as x → ∞, where Sis a sum �1 + ... + �� of random sizeand Mis a maximum of partial sums M� = maxn�� Sn. Heren, n = 1, 2, . . . , are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case whereis independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where E� > 0 and where the tail ofis comparable with or heavier than that of �, and obtain the asymptotics P{S� > x} ∼ EP{�1 > x} + P{� > x/E�} as x → ∞. 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{Sn > x}/P{�1 > x} which substantially improve Kesten's bound in the subclass Sof subexpo-