Uniform asymptotics for the tail probability of weighted sums with heavy tails

This paper studies the tail probability of weighted sums of the form Sigma(n)(i=1) c(i)X(i), where random variables Xi's are either independent or pairwise quasi-asymptotically independent with heavy tails. Using the idea of uniform long-tailedness, the uniform asymptotic equivalence of the tail probabilities of Sigma(n)(i=1) c(i)X(i), max(1 <= k <= n) Sigma(k)(i=1) c(i)X(i) and Sigma(n)(i=1) c(i)X(i)(+) is established, where Xi's are independent and follow the long-tailed distribution, and c(i)'s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of X's with dominated varying tails are obtained. An application to the ruin probability in a discrete-time insurance risk model is presented. (C) 2014 Elsevier B.V. All rights reserved.