On the maximum of randomly weighted sums with regularly varying tails

Abstract

Consider the randomly weighted sums Sn(θ) = ∑k=1 n θkXk, n = 1,2,..., where {Xk, k = 1, 2,...} is a sequence of independent real-valued random variables with common distribution F, whose right tail is regularly varying with exponent - α < 0, and {θk, k = 1,2,...} is a sequence of positive random variables, independent of {Xk, k = 1,2,...}. Under a suitable summability condition on the upper endpoints of θk, k = 1, 2,..., we prove that Pr(max1≤n<∞ Sn(θ) > x) ∼ F̄(x)∑k=1 ∞ Eθk α. © 2005 Elsevier B.V. All rights reserved.