On the tail behavior for randomly weighted sums of dependent random variables with its applications to risk measures

Abstract

This paper considers the asymptotic behavior for the tail probability of randomly weighted sum Sθ 2 = θ1X1+θ2X2, where X1, X2, θ1, and θ2 are non-negative dependent random variables with distributions F1, F2, G1, and G2, respectively. We obtain the tail-equivalence of P Sθ 2 > x and P(θ1X1 > x)+P(θ2X2 > x) as x → ∞ and some closure properties of distribution classes in three cases: (i). θ1, θ2 are bounded and F1, F2 are subexponential; (ii). θ1, θ2 satisfy the condition of Theorem 2.1 of Tang (2006) [33] and F1, F2 are subexponential with positive lower Matuszewska indices; (iii). θ1, θ2 satisfy the condition of Theorem 3.3 (iii) of Cline and Samorodnitsky (1994) [12] and F1, F2 are long-tailed and dominatedlyvarying- tailed. Furthermore, when F1 and F2 are regularly-varying-tailed, a more transparent result is established and applied to obtain asymptotic results for risk measures. Some numerical studies are conducted to check the accuracy of the obtained results.