View article

We study conditions under which P{S τ > x} ∼ P {M 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 = max n≤τ S n . Here, ξ n , 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{S 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 …