Convex geometry of max-stable distributions

TLDR: In this article, it was shown that max-stable random vectors with unit Frechet marginals are in one to one correspondence with convex sets K in [0, ∞ )====== d called max-zonoids, which can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or as the selection expectation of a random crosspolytope whose distribution is controlled by the spectral measure of the max-safe vector.

Abstract: It is shown that max-stable random vectors in [0, ∞ ) d with unit Frechet marginals are in one to one correspondence with convex sets K in [0, ∞ ) d 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 P ξ ≤ x of a max-stable random vector ξ with unit Frechet marginals is determined by the norm of the inverse to x, where all possible norms are given by the support functions of (normalised) 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.