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Let {X _k , 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 {X _k , 1 ≤ k ≤ n} and satisfying a ≤ θ _k ≤ b for some 0 < a ≤ b < ∞ for all 1 ≤ k ≤ n. This paper proves that the asymptotic relations P (max_{1 ≤ m ≤ n} ∑ _k=1 ^m θ _k X _k > x) ∼ P (sum _k=1 ⁿ θ _k X _k > x) ∼ sum _k=1 ⁿ P (θ _k X _k > x) hold as x → ∞. In doing so, no any assumption is made on the dependence structure of the sequence {θ _k , 1 ≤ k ≤ n}. An application to ruin theory is proposed.