Weighted sums of subexponential random variables and their maxima

Abstract

Let {Xk, k = 1, 2,...} be a sequence of independent random variables with common subexponential distribution F, and let {wk, k = 1, 2,...} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1 n wkXk and the maximum of weighted sums max1≤m≤n ∑k=1 m wkXk, subject to the requirement that they should hold uniformly for n = 1, 2,.... Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss Xk during the kth year, with a discount or inflation factor wk. © Applied Probability Trust 2005.