Operator Tail Densities of Multivariate Copulas

TL;DR

This paper introduces the concept of operator tail densities for copulas, providing a new way to analyze tail dependence in multivariate distributions through operator regular variation.

Contribution

It defines operator tail densities for copulas and characterizes operator-regularly-varying densities via copula tail densities and marginal regular variation.

Findings

Operator tail densities can be explicitly characterized for certain copulas.

Liouville copulas admit an explicit operator tail-dependence function despite lacking a closed form.

The approach decomposes multivariate tail behavior into copula and marginal components.

Abstract

Operator regular variation of a multivariate distribution can be decomposed into the operator tail dependence of the underlying copula and the regular variation of the univariate marginals. In this paper, we introduce operator tail densities for copulas and show that an operator-regularly-varying density can be characterized through the operator tail density of its copula together with the marginal regular variation. As an example, we demonstrate that although a Liouville copula is not available in closed form, it nevertheless admits an explicit operator tail-dependence function.