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 ${mathbb{N}^{n}_{0}}$ .