On the Tail Behavior of Sums of Dependent Risks

Abstract

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

Asymptotic Behavior of Extremal Events for Aggregate Dependent Random Variables

“…In this paper, we establish analogous results for an aggregate loss of the form g(X 1 , . The main results generalize those in Wüthrich [16], Alink et al [2], Barbe et al [4], and Embrechts et al [9]. .…”

“…Proof: From Barbe et al [4], it is known that h α (w)/n is a probability density function on S n−1 for all α > 0. Denote by H α the corresponding distribution function.…”

“…Under Model 3.1 with n ≥ 2, Barbe et al [4], Embrechts et al [9] and Chen et al [6] have obtained the monotonicities and boundary values of the limiting constants q F g (α, β),…”

“…For example, choose g 4 (4) In Theorem 4.1, sufficient conditions are given under which q F g (α, β) and q W g (α, β) possess monotonicities with respect to β.…”

“…For n > 2, Barbe, Fougères and Genest [4] and Embrechts, Nešlehová and Wüthrich [9] gave the properties of the limiting constant q F n for the Fréchet case, and Chen et al [6] investigate properties of the limiting constants q W n and q G n for the Weibull and Gumbel cases. In this paper, we establish analogous results for an aggregate loss of the form g(X 1 , .…”