Conditional tail expectation of randomly weighted sums with heavy-tailed distributions

We consider the tail behavior of the conditional tail expectation E(Snθ∣Snθ>xq) when q↑1q↑1. Here Snθ=∑i=1nθiXi and xq=VaRq(Snθ)=inf{y∈R:P(Snθ⩽y)⩾q}. We are interested in the case when the primary random variables X1,X2,…,XnX1,X2,…,Xn are real-valued and regularly varying, while the random weights θ1,θ2,…,θnθ1,θ2,…,θn are nonnegative and not degenerate at zero. We suppose that random vectors (X1,θ1),(X2,θ2),…(Xn,θn)(X1,θ1),(X2,θ2),…(Xn,θn) are independent, while XkXk and θkθk follow a certain dependence structure. We also present the related asymptotic results, some of which hold if distribution functions of X1,X2,…,XnX1,X2,…,Xn are long tailed and dominatingly varying.