Uniform approximation of the tail probability of weighted sums of subexponential random variables

Abstract

For subexponential random variables Xi, 1 ⩽ i ⩽ n, the uniform asymptotic result is established for 0 < an(x) ⩽ ci ⩽ bn(x), 1 ⩽ i ⩽ n, where an(x) decreases to 0 and b(x) goes to infinity as x tends to infinity. It extends the current result when the subexponential random variables Xi’s are independent and gives the first uniform asymptotic result when Xi’s are dependent.