The realization problem for tail correlation functions

TL;DR

This paper investigates the mathematical characterization of tail correlation functions (TCFs) for stochastic processes, providing a complete description of their structure, especially for finite index sets, and revealing their complex geometric properties.

Contribution

It offers a complete characterization of TCFs via affine inequalities and describes their geometric structure, especially for finite index sets, advancing understanding of tail dependence modeling.

Findings

TCFs coincide with those from a subclass of max-stable processes.

The set of TCFs forms a convex polytope for finite T.

Necessary and sufficient conditions for TCFs are identified up to |T|=6.

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.