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

We consider the tail behavior of the conditional tail expectation E(S-n(theta) vertical bar S-n(theta) > x(q)) when q up arrow 1. Here S-n(theta) = Sigma(n)(i=1) theta X-i(i) and x(q) = VaR(q)(S-n(theta)) = inf{y is an element of R : P(S-n(theta) <= y) >= q}. We are interested in the case when the primary random variables X-1, X-2, ..., X-n are realvalued and regularly varying, while the random weights theta(1), theta(2), ..., theta(n) are nonnegative and not degenerate at zero. We suppose that random vectors (X-1, theta(1)), (X-2, theta(2)), ... (X-n, theta(n)) are independent, while X-k and theta(k) follow a certain dependence structure. We also present the related asymptotic results, some of which hold if distribution functions of X-1, X-2, ..., X-n are long tailed and dominatingly varying. (C) 2015 Elsevier B.V. All rights reserved.