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ASYMPTOTIC BEHAVIOR OF EXTREMAL EVENTS FOR AGGREGATE DEPENDENT RANDOM VARIABLES

Published online by Cambridge University Press:  13 August 2013

Die Chen ,
Tiantian Mao  and
Taizhong Hu
Die Chen
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China; Department of School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China E-mail: chendie@mail.ustc.edu.cn
Tiantian Mao
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mails: tmao@ustc.edu.cn; thu@ustc.edu.cn
Taizhong Hu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mails: tmao@ustc.edu.cn; thu@ustc.edu.cn
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Abstract

Consider a portfolio of n identically distributed risks X1, …, Xn with dependence structure modelled by an Archimedean survival copula. It is known that the probability of a large aggregate loss of $\sum\nolimits_{i=1}^{n} X_{i}$ is in proportion to the probability of a large individual loss of X1. The proportionality factor depends on the dependence strength and the tail behavior of the individual risk. In this paper, we establish analogous results for an aggregate loss of the form g(X1, …, Xn) under the more general model in which the Xi's have different but tail-equivalent distributions and the copula remains unchanged, where g is a homogeneous function of order 1. Properties of these factors are studied, and asymptotic Value-at-Risk behaviors of functions of dependent risks are also given. The main results generalize those in Wüthrich [16], Alink, Löwe, and Wüthrich [2], Barbe, Fougères, and Genest [4], and Embrechts, Nešlehová, and Wüthrich [9].

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 27 , Issue 4 , October 2013 , pp. 507 - 531
Copyright
Copyright © Cambridge University Press 2013 

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