{
  "id": "math/0603423",
  "version": "v3",
  "published": "2006-03-17T11:23:42.000Z",
  "updated": "2007-10-29T12:12:26.000Z",
  "title": "Convex geometry of max-stable distributions",
  "authors": [
    "Ilya Molchanov"
  ],
  "comment": "25 pages. Revised version",
  "categories": [
    "math.PR"
  ],
  "abstract": "It is shown that max-stable random vectors in $[0,\\infty)^d$ with unit Fr\\'echet 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 $\\Prob{\\xi\\leq x}$ of a max-stable random vector $\\xi$ with unit Fr\\'echet 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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-10-29T12:12:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "60D05"
    ],
    "keywords": [
      "max-stable random vector",
      "max-stable distributions",
      "unit frechet marginals",
      "max-zonoids",
      "convex geometry approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3423M"
    }
  }
}