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

Zhangting Chen () and Dongya Cheng ()
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Zhangting Chen: Soochow University
Dongya Cheng: Soochow University

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 4, 1-27

Abstract: Abstract 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.

Keywords: Randomly weighted sum; Dependent random variable; Subexponential distribution; Risk measure; Numerical study; Primary 62E20; Secondary 60E05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-024-10118-6

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