From Geometry of the Limit Set to Extremal Dependence Properties of Light-tailed Distributions

Sample clouds of multivariate data points from light-tailed distributions can often be scaled to converge onto a deterministic set as the sample size tends to infinity. It turns out that the shape of this limit set can be related to a number of extremal dependence properties of the underlying distribution. In this talk, I will present several simple relations, and illustrate how they can be used to replace frequently cumbersome or intractable analytical computations. As an application from the area of finance, I will discuss a new approach to quantifying the effect of risk diversification, which arises from forming a portfolio of risky assets whose behavior is determined by a multivariate probability density. The attention will be restricted to the class of densities whose level sets are all scaled copies of a given set. Exploiting the geometric structure of this common shape we are able to measure the effect of risk aggregation on extremes as well as to quantify the impact of dimension on diversification.