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

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发表日期：

2024年11月5日

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Zhangting Chen,
Dongya Cheng

摘要

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 $ $x \to \infty$ 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.