The realization problem for tail correlation functions

摘要: For a stochastic process {Xt}t∈T with identical one-dimensional margins and upper endpoint τup its tail correlation function (TCF) is defined through \(\chi ^{(X)}(s,t) = \lim _{\tau \to \tau _{\text {up}}} P(X_{s} > \,\mid \, X_{t} )\). It popular bivariate summary measure that has been frequently used in the literature order to assess dependence. In this article, we study realization problem. We show set of all TCFs on T×T coincides stemming from subclass max-stable processes can be completely characterized by system affine inequalities. Basic closure properties regularity implications continuity χ are derived. If T finite, forms convex polytope \(\lvert \rvert \times \lvert \) matrices. Several general results reveal complex geometric structure. Up 6\) reduced necessary sufficient conditions for being TCF determined. None these will become obsolete as \geq 3\) grows.