2006/03/17 by Ilya Molchanov, Molchanov, Ilya · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60D05 #60G70 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Risk and Portfolio Optimization #Stochastic processes and financial applications #math.PR #msc:60D05 #msc:60G70
paper · pdf · doi:10.48550/arxiv.math/0603423
25 pages. Revised version
openalex publication_date 2006/03/17 · arxiv created 2007/10/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is shown that max-stable random vectors in [0,∞)d with unit Fréchet 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 \Probξ≤ x of a max-stable random vector ξ with unit Fréchet marginals is determined by the norm of the inverse to x, 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.