Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

High Dimensional Probability VIIIA Probabilistic Characterization of Negative Definite Functions

High Dimensional Probability VIII: A Probabilistic Characterization of Negative Definite Functions [It is proved that a continuous function f on ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {R}^n$$ \end{document} is negative definite if and only if it is polynomially bounded and satisfies the inequality 𝔼f(X−Y)≤𝔼f(X+Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {E} f(X-Y)\le \mathbb {E} f(X+Y)$$ \end{document} for all i.i.d. random vectors X and Y  in ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {R}^n$$ \end{document}. The proof uses Fourier transforms of tempered distributions. The “only if” part has been proved earlier by Lifshits et al. (A probabilistic inequality related to negative definite functions. Progress in probability, vol. 66 (Springer, Basel, 2013), pp. 73–80).] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

High Dimensional Probability VIIIA Probabilistic Characterization of Negative Definite Functions

Part of the Progress in Probability Book Series (volume 74)
Editors: Gozlan, Nathael; Latała, Rafał; Lounici, Karim; Madiman, Mokshay
Gao, Fuchang
High Dimensional Probability VIII — Nov 27, 2019
Download PDF
12 pages

Loading next page...
 
/lp/springer-journals/high-dimensional-probability-viii-a-probabilistic-characterization-of-qq4Y2P7xzQ

References (12)

Publisher
Springer International Publishing
Copyright
© Springer Nature Switzerland AG 2019
ISBN
978-3-030-26390-4
Pages
41 –53
DOI
10.1007/978-3-030-26391-1_5
Publisher site
See Chapter on Publisher Site

Abstract

[It is proved that a continuous function f on ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {R}^n$$ \end{document} is negative definite if and only if it is polynomially bounded and satisfies the inequality 𝔼f(X−Y)≤𝔼f(X+Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {E} f(X-Y)\le \mathbb {E} f(X+Y)$$ \end{document} for all i.i.d. random vectors X and Y  in ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {R}^n$$ \end{document}. The proof uses Fourier transforms of tempered distributions. The “only if” part has been proved earlier by Lifshits et al. (A probabilistic inequality related to negative definite functions. Progress in probability, vol. 66 (Springer, Basel, 2013), pp. 73–80).]

Published: Nov 27, 2019

Keywords: Negative definite function; Lévy–Khintchine representation; Fourier inversion theorem; Polynomially bounded; Primary: 60E15; 42A82; Secondary: 42B10; 60E10

There are no references for this article.