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Convex geometry of max-stable distributions

Ilya Molchanov

TL;DR

The paper develops a convex-geometry framework for max-stable distributions, showing a one-to-one correspondence between unit-Fréchet max-stable laws and max-zonoids, which are limits of cross-polytope sums controlled by a spectral measure. It characterizes the corresponding norms, tail-dependence structures, and copulas via support functions of dependency sets, and establishes a wide array of geometric operations (scaling, projection, Minkowski sums, power means, duality) that translate into probabilistic constructions. A key contribution is the identification of precise conditions under which a convex set is a max-zonoid, and the elucidation of how spectral measures, extremal coefficients, and copula properties are encoded in the geometry of K and its polar. The finite- and infinite-dimensional developments provide new tools to construct, compare, and analyze dependency structures in multivariate and functional extreme-value modeling, with potential applications to copulas and dependence quantification in higher dimensions.

Abstract

It is shown that max-stable random vectors in $[0,\infty)^d$ with unit Fréchet marginals are in one to one correspondence with convex sets $K$ in $[0,\infty)^d$ 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 $\Prob{ξ\leq x}$ 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.

Convex geometry of max-stable distributions

TL;DR

The paper develops a convex-geometry framework for max-stable distributions, showing a one-to-one correspondence between unit-Fréchet max-stable laws and max-zonoids, which are limits of cross-polytope sums controlled by a spectral measure. It characterizes the corresponding norms, tail-dependence structures, and copulas via support functions of dependency sets, and establishes a wide array of geometric operations (scaling, projection, Minkowski sums, power means, duality) that translate into probabilistic constructions. A key contribution is the identification of precise conditions under which a convex set is a max-zonoid, and the elucidation of how spectral measures, extremal coefficients, and copula properties are encoded in the geometry of K and its polar. The finite- and infinite-dimensional developments provide new tools to construct, compare, and analyze dependency structures in multivariate and functional extreme-value modeling, with potential applications to copulas and dependence quantification in higher dimensions.

Abstract

It is shown that max-stable random vectors in with unit Fréchet marginals are in one to one correspondence with convex sets in 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.

Paper Structure

This paper contains 16 sections, 17 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Dependency sets and max-zonoids
  3. Norms associated with max-stable distributions
  4. Spectral measures
  5. Copulas and association
  6. Complete alternation and extremal coefficients
  7. Operations with dependency sets
  8. Rescaling.
  9. Projection.
  10. Cartesian product.
  11. Minkowski sum.
  12. Power sums.
  13. Minkowski difference.
  14. Convex hull and intersection.
  15. Duality.
  16. ...and 1 more sections

Key Result

Theorem 2.1

A random vector $\xi$ is max-stable with unit Fréchet marginals if and only if its cumulative distribution function $F(x)=\mathbf{P}\{\xi\leq x\}$ satisfies for a constant $c>0$ and a random vector $\eta\in{\mathbb{S}_+}$ such that $c\mathop{\mathrm{\bf E}}\nolimits\eta=(1,\dots,1)$.

Theorems & Definitions (38)

  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • proof
  • ...and 28 more