Asymptotics of randomly stopped sums in the presence of heavy tails

We study conditions under which P_τ >x\ ∼ P_τ > x\ ∼ EτP\ξ₁ > x\ as x → ∞, where S_τ is a sum ξ₁ + ⋯ + ξ_τ of random size $τ$ and M_τ is a maximum of partial sums M_τ = 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_τ > 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 Pₙ > x\ / P\ξ₁ > x\ which substantially improve Kesten’s bound in the subclass S* of subexponential distributions.