Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions

@article{McNeil2009MultivariateAC,
  title={Multivariate Archimedean copulas, \$d\$-monotone functions and \$\ell\_1\$-norm symmetric distributions},
  author={Alexander J. McNeil and Johanna Nevslehov'a},
  journal={arXiv: Statistics Theory},
  year={2009}
}
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 $\ell_1$-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… 

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