Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation

Abstract

ABSTRACTLet Xk, 1 ⩽ k ⩽ n, be n real-valued random variables, and θk, 1 ⩽ k ⩽ n, be another n non negative not-degenerate at zero random variables. Assume that random pairs (X1, θ1), …, (Xn, θn) are mutually independent, while each pair (Xk, θk) follows a wide type of dependence structure. Consider the randomly weighted sum Sθn = ∑k = 1nθkXk. In this paper, the tail asymptotics for Sθn in the case where Xk, 1 ⩽ k ⩽ n, belong to some heavy-tailed subclasses are firstly investigated. Then, as an application, we consider the tail behavior of the conditional tail expectation as q↑1, where . Under some technical conditions, the asymptotic estimate for the right tail of conditional tail expectation is also derived. The obtained results extend some existing ones in the literature.