Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {Xk, 1 ≤ k ≤ n} be n independent and real-valued random variables with common subexponential distribution function, and let {θk, 1 ≤ k ≤ n} be other n random variables independent of {Xk, 1 ≤ k ≤ n} and satisfying a ≤ θk ≤ b for some 0 < a ≤ b < ∞ for all 1 ≤ k ≤ n. This paper proves that the asymptotic relations P (max1 ≤ m ≤ n ∑k=1m θkXk > x) ∼ P (sumk=1n θkXk > x) ∼ sumk=1nP (θkXk > x) hold as x → ∞. In doing so, no any assumption is made on the dependence structure of the sequence {θk, 1 ≤ k ≤ n}. An application to ruin theory is proposed.