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

Let X_k, 1 ⩽ k ⩽ n, be n real-valued random variables, and θ_k, 1 ⩽ k ⩽ n, be another n non negative not-degenerate at zero random variables. Assume that random pairs (X₁, θ₁), …, (X_n, θ_n) are mutually independent, while each pair (X_k, θ_k) follows a wide type of dependence structure. Consider the randomly weighted sum S^θ_n = ∑_{k = 1}ⁿθ_kX_k. In this paper, the tail asymptotics for S^θ_n in the case where X_k, 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 <inline-graphic></inline-graphic> as q↑1, where <inline-graphic></inline-graphic>. 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. [ABSTRACT FROM AUTHOR]