On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform
share
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 𝒞_ar^d 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 𝒞_ar^d 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 μ_C(L_t) and the Kendall distribution function F_K^d 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 μ_C(L_t) and F_K^d in terms of γ and to characterize pointwise convergence in 𝒞_ar^d by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of γ directly carry over to the corresponding copula C_γ∈𝒞_ar^d. 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 𝒞_ar^d 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.
READ FULL TEXT