# 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
12 pages

/lp/springer-journals/high-dimensional-probability-viii-a-probabilistic-characterization-of-qq4Y2P7xzQ

# References (12)

Publisher
Springer International Publishing
© 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