广义负相依重尾随机变量和及其最大值尾概率的渐近性

Asymptotics for Tail Probabilities of the Sum and Its Maximum of Extended Negatively Dependent and Heavy-Tailed Random Variables

摘要: 假设X_1,X_2,\ldots,X_n是一列具有广义负相依结构的随机变量(r.v.s.), 分别具有分布F_1,F_2,\ldots,F_n.假设S_n:=X_1+X_2+\cdots+X_n.本文分别在三类重尾分布族下得到了如下量之间的渐近关系: \pr(S_n>x),\pr(\max\X_1,X_2,\ldots,X_n\>x), \pr(\max\S_1,S_2,\ldots,S_n\>x)~和~\tsm_k=1^n\pr(X_k>x). 在此基础上,本文还探讨了随机加权和最大值尾概率的渐近性质,并运用蒙特卡洛~(CMC)~数值模拟验证了其有效性. 最后,本文将得到的主要结果应用到了一个带有保险风险与金融风险的离散时间风险模型,得到了有限时间破产概率的渐近性.

Abstract: Let X_1,X_2,\ldots,X_n be a sequence of extended negatively dependent random variables with distributions F_1,F_2,\ldots,F_n,respectively. Denote by S_n=X_1+X_2+\cdots+X_n. This paper establishes the asymptotic relationship for the quantities \pr(S_n>x), \pr(\max\X_1,X_2, \ldots,X_n\>x), \pr(\max\S_1,S_2, \ldots,S_n\>x) and \tsm_k=1^n\pr(X_k>x) in the three heavy-tailed cases. Based on this, this paper also investigates the asymptotics for the tail probability of the maximum of randomly weighted sums, and checks its accuracy via Monte Carlo simulations. Finally, as an application to the discrete-time risk model with insurance and financial risks, the asymptotic estimate for the finite-time ruin probability is derived.