Decompositions of dependence for high-dimensional extremes

Abstract

We propose two decompositions that help to summarize and describe high-dimensional tail dependence within the framework of regular variation.We use a transformation to define a vector space on the positive orthant and show that transformed-linear operations applied to egularlyvarying random vectors preserve regular variation.We summarize tail dependence via a matrix of pairwise tail dependence metrics that is positive semidefinite; eigendecomposition allows one to interpret tail dependence in terms of the resulting eigenbasis. This matrix is completely positive, and one can easily construct regularly-varying random vectors that share the same pairwise tail dependencies.We illustrate our methods with Swiss rainfall and financial returns data.