Operator Tail Densities of Multivariate Copulas
Haijun Li
TL;DR
The paper develops the theory of operator tail densities for copulas to analyze multivariate regular variation without requiring tail-equivalent margins. It shows that an operator-regularly-varying density can be characterized by the copula's operator tail density together with univariate marginal regular variation, and provides an explicit transformation formula linking joint and copula tails under diagonal operator norming. A key result is the decomposition formula $\lambda(w)=\lambda_C((w_1^{-\alpha_1},\dots,w_d^{-\alpha_d});(1,\dots,1))|J(w_1^{-\alpha_1},\dots,w_d^{-\alpha_d})|$ with $\alpha_i=\rho/\lambda_i$, which connects copula-scale tail structure to margins. The Liouville copula example demonstrates that explicit operator tail densities can exist for copulas even when the joint density is not in closed form, highlighting the practical value of the approach for capturing scale-invariant tail dependence in complex multivariate models.
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.