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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 $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with $d \geq 3$ and proves the afore-mentioned implication with respect to conditioning on the first $d-1$ coordinates. Several proper\-ties equivalent to pointwise convergence in $\mathcal{C}_{ar}^d$ 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 $\mu_C(L_t)$ and the Kendall distribution function $F_K^d$ as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators $\psi$ of $d$-dimensional Archimedean copulas from the perspective of their so-called Williamson measures $\gamma$ on $(0,\infty)$ is then shown to allow not only to derive surprisingly simple expressions for $\mu_C(L_t)$ and $F_K^d$ in terms of $\gamma$ and to characterize pointwise convergence in $\mathcal{C}_{ar}^d$ by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of $\gamma$ directly carry over to the corresponding copula $C_\gamma \in \mathcal{C}_{ar}^d$. 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 $\mathcal{C}_{ar}^d$ 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.

Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance - in fact, given samples of a copula $C$ the corresponding empirical copulas do not converge weakly conditional to $C$ with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and weak conditional convergence can even be proved to be equivalent. Moreover, it can be shown that every copula $C$ is the weak conditional limit of a sequence of checkerboard copulas. After proving these three main results and pointing out some consequences we sketch some implications for two recently introduced dependence measures and for the nonparametric estimation of Archimedean and Extreme Value copulas., Comment: 23 pages, 9 figures