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

Let 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 E(Snθ|Snθ>xq)$\mathbb {E}(S_n^\theta |S_n^\theta >x_q)$ as q↑1, where xq=VaRq(Snθ)=inf{y∈R:P(Snθ≤y)≥q}$x_q=\mbox{VaR}_q(S_n^\theta )=\inf \lbrace y\in \mathbb {R}: \mathbb {P}(S_n^\theta \le y)\ge q\rbrace$. 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.