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The realization problem for tail correlation functions

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Abstract

For a stochastic process {X t } t∈T with identical one-dimensional margins and upper endpoint τ 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×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 χ are derived. If T is finite, the set of TCFs on T×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.

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Authors and Affiliations

  1. Institute for Mathematical Stochastics, Faculty of Mathematics and Computer Science, University of Göttingen, D-37077, Göttingen, Germany

    Ulf-Rainer Fiebig

  2. Institute of Mathematics, University of Mannheim, D-68131, Mannheim, Germany

    Kirstin Strokorb & Martin Schlather

Authors
  1. Ulf-Rainer Fiebig
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  2. Kirstin Strokorb
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  3. Martin Schlather
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Correspondence to Kirstin Strokorb.

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Fiebig, UR., Strokorb, K. & Schlather, M. The realization problem for tail correlation functions. Extremes 20, 121–168 (2017). https://doi.org/10.1007/s10687-016-0250-8

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