A note on the asymptotics for the randomly stopped weighted sums

Abstract

Let {X_i , i ⩾ 1} be a sequence of identically distributed real-valued random variables with common distribution F_X; let {θ_i , i ⩾ 1} be a sequence of identically distributed, nonnegative and nondegenerate at zero random variables; and let τ be a positive integer-valued counting random variable. Assume that {X_i , i ⩾ 1}, {θ_i , i ⩾ 1} and τ are mutually independent. In the presence of heavy-tailed X_i's, this paper investigates the asymptotic tail behavior for the maximum of randomly weighted sums M_τ = max_{1 ⩽ k ⩽ τ} ∑^k_{i = 1}θ_i X_i under the condition that {θ_i , i ⩾ 1} satisfy a general dependence structure.