Anja Janssen: Metric embeddings of tail correlation matrices

Abstract

The extremal dependence between components of a random vector is important for the assessment of risks determined by several factors, for example in the evaluation of risk measures of financial or insurance portfolios. One popular bivariate measure of tail risk is the matrix of tail correlation coefficients. The entries of this matrix are closely related to a useful distance measure on the space of Fréchet(1)-random variables, named spectral tail distance. We analyze the properties of the related metric and show that it is L^1- and l_1-embeddable, which allows us to better understand its structure. In particular, a given embedding of the spectral tail distance can be related to a certain max-stable random vector realizing a given tail correlation matrix. That way, our approach allows us to conclude about the algorithmic complexity of the decision problem whether a given matrix is a tail correlation matrix or not.  This is joint work with Sebastian Neblung (University of Hamburg) and Stilian Stoev (University of Michigan)