Title: Revisiting the Williamson transform in the context of multivariate Archimedean copulas
Authors: Nicolas Dietrich - Universität Salzburg (Austria) [presenting]
Wolfgang Trutschnig - University of Salzburg (Austria)
Thimo Kasper - University of Salzburg (Austria)
Abstract: A very recent result states that within the family of all $d$-dimensional Archimedean copulas, standard pointwise convergence implies $d-1$ weak conditional convergence (that is, weak convergence of almost all $(d-1)$-Markov kernels), and it is well-known from the literature that pointwise convergence within the family of multivariate Archimedean copulas is equivalent to the convergence of the corresponding normalized generators. We view multivariate Archimedean copulas via the Williamson transform, i.e., we study probability measures on $(0,\infty)$ whose corresponding Williamson transform coincides with the considered generators. Using this handy interrelation, it is not only possible to derive alternative handy formulas for the mass of level sets of the copulas but also to prove that both afore-mentioned notions of convergence may fully be characterized in terms of weak convergence of the probability measures on $(0,\infty)$. Furthermore, even singularity properties of the Archimedian copulas may directly be derived from the probability measures on $(0,\infty)$.