Higher order monotonic (multi-) sequences and their extreme points

Functions on the half-line which are non-negative and decreasing of a higher order have a long tradition. When normalized they form a simplex whose extreme points are well-known. For functions on ${\mathbb{N}_{0} = \{0, 1, 2, . . .\}}$ the situation is different. Since an n-monotone sequence is in general not the restriction of an n-monotone function on ${\mathbb{R}_{+}}$ (apart from n = 1 and n = 2), it is not even clear at the beginning if the normalized n-monotone sequences form a simplex. We will show in this paper that this is actually true, and we determine their extreme points. A corresponding result will also be proved for multi-sequences. The main ingredient in the proof will be a relatively new characterization of so-called survival functions of probability measures on (subsets of) ${\mathbb{R}^n}$ , in this case on ${\mathbb{N}^{n}_{0}}$ .     

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.