Asymptotics of randomly stopped sums in the presence of heavy tails

We study conditions under which P{Sτ > x} ∼ P{M τ > x} ∼ EτP{ξ1 > x} as x→∞, where Sτ is a sum ξ1+ ⋯ ξτ of random size τ and Mτ is a maximum of partial sums Mτ = maxn≤τ Sn. Here, ξn, n = 1, 2,. . ., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where τ is independent of the summands; also, in a particular situation, we deal with a stopping time. We also consider the case where Eξ > 0 and where the tail of τ is comparable with, or heavier than, that of ξ, and obtain the asymptotics P{Sτ > x} ∼ EτP{ξ1 > x} + P{τ > x/Eξ} as x→∞. 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{Sn > x}/P{ξ1 > x} which substantially improve Kesten's bound in the subclass S* of subexponential distributions. © 2010 ISI/BS.