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

Let {Xk,1≤k≤n}{Xk,1≤k≤n} be nn independent and real-valued random variables with common subexponential distribution function FF. 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 Sn=∑k=1nckXk and we find an interval [u(x),v(x)][u(x),v(x)] with u(x)↓0u(x)↓0 and v(x)↑∞v(x)↑∞ as x→∞x→∞ such that the asymptotic relation P(Sn>x)∼∑k=1nP(ckXk>x) holds uniformly for all weights ck,1≤k≤nck,1≤k≤n, taking values from this interval.