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

Abstract

This paper studies the tail probability of weighted sums of the form n i=1 ciXi, where random variables Xi's are either independent or pairwise quasi-asymptotical independent with heavy tails. Using h-insensitive function, the uniform asymptotic equivalence of the tail probabilities ofwhere Xi's are independent and follow the long-tailed distribution, and ci's take value in a broad interval. Some further uniform asymptotic results for the weighted sums of Xi's with dominated varying tails are obtained. An application to the ruin probability in a discrete-time insurance risk model is presented. MSC: 41A60; 62P05; 62E20; 91B30