ASYMPTOTIC BEHAVIOR OF EXTREMAL EVENTS FOR AGGREGATE DEPENDENT RANDOM VARIABLES

摘要

Consider a portfolio of n identically distributed risks X_1,..., X_n with dependence structure modelled by an Archimedean survival copula. It is known that the probability of a large aggregate loss of ∑_(i=1)~n X_i is proportion to the probability of a large individual loss of X_1. 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(X_1,..., X_n) under the more general model in which the X_i'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 Wuethrich [16], Alink, Loewe, and Wuethrich [2], Barbe, Fougeres, and Genest [4], and Embrechts, Neslehova, and Wiithrich [9].