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The realization problem for tail correlation functions. (English) Zbl 1369.60036

Summary: For a stochastic process \(\{X_{t}\}_{t\in T}\) with identical one-dimensional margins and upper endpoint \(\tau_{\mathrm{up}}\) its tail correlation function (TCF) is defined through \(\chi ^{(X)}(s,t) = \lim _{\tau \rightarrow \tau _{\mathrm {up}}} P(X_{s} > \tau \,\mid \, X_{t} > \tau )\). 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 \(| T | \times | T | \) matrices. Several general results reveal its complex geometric structure. Up to \(| T | = 6\) a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as \(| T | \geq 3\) grows.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60G52 Stable stochastic processes
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Software:

R; polymake
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References:

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