Convex geometry of max-stable distributions

Ilya Molchanov

Paper

Convex geometry of max-stable distributions

Abstract

It is shown that max-stable random vectors in

with unit Fréchet marginals are in one to one correspondence with convex sets

called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function

of a max-stable random vector

with unit Fréchet marginals is determined by the norm of the inverse to

, where all possible norms are given by the support functions of max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided. The convex geometry approach makes it possible to introduce new operations with max-stable random vectors.