Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric distributions

It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of d-dimensional ℓ1-norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189–207] in an analogous manner to the well-known Bernstein–Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean…