{
  "id": "1405.6876",
  "version": "v2",
  "published": "2014-05-27T11:58:16.000Z",
  "updated": "2016-03-24T17:00:59.000Z",
  "title": "The realization problem for tail correlation functions",
  "authors": [
    "Ulf-Rainer Fiebig",
    "Kirstin Strokorb",
    "Martin Schlather"
  ],
  "comment": "42 pages, 7 Tables",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a stochastic process $\\{X_t\\}_{t \\in T}$ with identical one-dimensional margins and upper endpoint $\\tau_{\\text{up}}$ its tail correlation function (TCF) is defined through $\\chi^{(X)}(s,t) = \\lim_{\\tau \\to \\tau_{\\text{up}}} P(X_s > \\tau \\,\\mid\\, X_t > \\tau )$. It is a popular bivariate summary measure that has been frequently used in the literature in order to assess tail dependence. In this article, we study its realization problem. We show that the set of all TCFs on $T \\times T$ coincides with the set of TCFs stemming from a subclass of max-stable processes and can be completely characterized by a system of affine inequalities. Basic closure properties of the set of TCFs and regularity implications of the continuity of $\\chi$ are derived. If $T$ is finite, the set of TCFs on $T \\times T$ forms a convex polytope of $\\lvert T \\rvert \\times \\lvert T \\rvert$ matrices. Several general results reveal its complex geometric structure. Up to $\\lvert T \\rvert = 6$ a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as $\\lvert T \\rvert\\geq 3$ grows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-27T11:58:16.000Z",
      "abstract": "Let $X=\\{X_t\\}_{t \\in T}$ be a stochastic process on an arbitrary index set $T$ with identical marginal distributions and upper endpoint $\\tau_{\\text{upper}}$. The tail correlation function (TCF) $\\chi$ of $X$ is defined through $\\chi(s,t) = \\lim_{\\tau \\to \\tau_{\\text{upper}}} P(X_s > \\tau \\,\\mid\\, X_t > \\tau )$ for $s,t \\in T$, provided the limit exists. We show that the set of all TCFs on $T \\times T$ coincides with the set of TCFs stemming from a subclass of max-stable processes. It can be completely characterized by finite-dimensional inequalities. If $T$ is finite, the set of TCFs on $T \\times T$ forms a convex polytope of $\\lvert T \\rvert \\times \\lvert T \\rvert$ matrices. Up to $\\lvert T \\rvert = 6$ its vertices and facet inducing inequalities are computed, the latter forming a reduced system of necessary and sufficient conditions for being a TCF. None of these conditions will become obsolete as $\\lvert T \\rvert$ grows.",
      "comment": "34 pages, 7 Tables",
      "journal": null,
      "doi": null,
      "authors": [
        "Ulf Fiebig",
        "Kirstin Strokorb",
        "Martin Schlather"
      ]
    },
    {
      "version": "v2",
      "updated": "2016-03-24T17:00:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "15B51",
      "52B12",
      "52B05",
      "05-04"
    ],
    "keywords": [
      "tail correlation function",
      "realization problem",
      "arbitrary index set",
      "identical marginal distributions",
      "upper endpoint"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6876F"
    }
  }
}