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

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 $${\mathbb{N}^{n}_{0}}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

Positivity , Volume 17 (2) – Mar 14, 2012

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Springer Basel AG
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0169-5
Publisher site
See Article on Publisher Site

Abstract

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

Journal

PositivitySpringer Journals

Published: Mar 14, 2012

References

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