On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform

Abstract

Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class Card of all d-dimensional Archimedean copulas with d≥3 and proves the afore-mentioned implication with respect to conditioning on the first d−1 coordinates. Several properties equivalent to pointwise convergence in Card are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses μC(Lt) and the Kendall distribution function FKd as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators ψ of d-dimensional Archimedean copulas from the perspective of their so-called Williamson measures γ on (0,∞) is then shown to allow not only to derive surprisingly simple expressions for μC(Lt) and FKd in terms of γ and to characterize pointwise convergence in Card by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of γ directly carry over to the corresponding copula Cγ∈Card. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular d-dimensional copulas is dense in Card and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.