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

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