On the maximum of randomly weighted sums with regularly varying tails

Consider the randomly weighted sums S-n(theta) = Sigma(n)(k=1) theta X-k(k), n = 1, 2,..., where (X-k, k = 1, 2....) is a sequence of independent real-valued random variables with common distribution F, whose right tail is regularly varying with exponent -alpha <0, and {theta(k), k = 1, 2,...} is a sequence of positive random variables, independent of {X-k, k = 1, 2,...}. Under a suitable summability condition on the upper endpoints of theta(k), k = 1, 2,..., we prove that Pr(max(1 <= n<infinity) S-n(theta) > x)similar to(F) over bar (x)Sigma(infinity)(k=1) E theta(alpha)(k). (C) 2005 Elsevier B.V. All rights reserved.