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Extremes

, Volume 11, Issue 3, pp 235–259 | Cite as

Convex geometry of max-stable distributions

  • Ilya MolchanovEmail author
Article

Abstract

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 P ξ ≤ 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 (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.

Keywords

Copula Max-stable random vector Norm Cross-polytope Spectral measure Support function Zonoid 

AMS 2000 Subject Classifications

Primary—60G70 Secondary—52A21, 60D05 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Mathematical Statistics and Actuarial ScienceUniversity of BernBernSwitzerland

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