The realization problem for tail correlation functions

Abstract

For a stochastic process {X (t) } (taT) with identical one-dimensional margins and upper endpoint tau (up) its tail correlation function (TCF) is defined through . 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 TxT 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 TxT forms a convex polytope of matrices. Several general results reveal its complex geometric structure. Up to a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as grows.