On the Tail Behavior for Randomly Weighted Sums of Dependent Random Variables with its Applications to Risk Measures

This paper considers the asymptotic behavior for the tail probability of randomly weighted sum $$S_2^{\theta }=\theta _1X_1+\theta _2X_2$$ S 2 θ = θ 1 X 1 + θ 2 X 2 , where $$X_1$$ X 1 , $$X_2$$ X 2 , $$\theta _1$$ θ 1 , and $$\theta _2$$ θ 2 are non-negative dependent random variables with distributions $$F_1$$ F 1 , $$F_2$$ F 2 , $$G_1$$ G 1 , and $$G_2$$ G 2 , respectively. We obtain the tail-equivalence of $$P\left( S_2^{\theta }>x\right) $$ P S 2 θ > x and $$P(\theta _1X_1>x)+P(\theta _2X_2>x)$$ P ( θ 1 X 1 > x ) + P ( θ 2 X 2 > x ) as $$x\rightarrow \infty $$ x → ∞ and some closure properties of distribution classes in three cases: (i). $$\theta _1$$ θ 1 , $$\theta _2$$ θ 2 are bounded and $$F_1$$ F 1 , $$F_2$$ F 2 are subexponential; (ii). $$\theta _1$$ θ 1 , $$\theta _2$$ θ 2 satisfy the condition of Theorem 2.1 of Tang (Extremes 9(3):231–241 2006) and $$F_1$$ F 1 , $$F_2$$ F 2 are subexponential with positive lower Matuszewska indices; (iii). $$\theta _1$$ θ 1 , $$\theta _2$$ θ 2 satisfy the condition of Theorem 3.3 (iii) of Cline and Samorodnitsky (Stochastic Process and their Appl 49(1):75-98 1994) and $$F_1$$ F 1 , $$F_2$$ F 2 are long-tailed and dominatedly-varying-tailed. Furthermore, when $$F_1$$ F 1 and $$F_2$$ F 2 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.