A revision of Kimberling’s results — With an application to max-infinite divisibility of some Archimedean copulas

In his paper A probabilistic interpretation of complete monotonicity Kimberling (1974) proves several remarkable results connecting multivariate distribution functions and their marginals via completely monotone functions on the half-line. These have been taken up more recently in particular in connection with so-called Archimedean copulas; see for example Nelsen (2006). We present in this paper much shorter proofs of more general versions of the two main theorems in Kimberling (1974), and apply this to show the max-infinite divisibility of some known Archimedean copulas.