The uniform approximation of the tail probability of the randomly weighted sums of subexponential random variables

Abstract

Let { X k , 1 ≤ k ≤ n } be n independent and real-valued random variables with common subexponential distribution function F . Following the work of Tang and Tsitsiashvili [Tang, Q., Tsitsiashvili, G., 2003. Randomly weighted sums of subexponential random variables with application to ruin theory, Extremes 6, 171–188], we revisit the weighted sum S n = ∑ k = 1 n c k X k and we find an interval [ u ( x ) , v ( x ) ] with u ( x ) ↓ 0 and v ( x ) ↑ ∞ as x → ∞ such that the asymptotic relation P ( S n > x ) ∼ ∑ k = 1 n P ( c k X k > x ) holds uniformly for all weights c k , 1 ≤ k ≤ n , taking values from this interval.