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(1991)
Functional Analysis International Series in Pure and Applied Mathematics, 2nd edn (McGraw-Hill
(2001)
Pseudo Differential Operators and Markov Processes Fourier Analysis and Semi-groups, vol. I (Imperial
I. Gel'fand, N. Vilenkin (1964)
Generalized Random Processes
I. Schoenberg (1938)
Metric spaces and positive definite functionsTransactions of the American Mathematical Society, 44
M. Lifshits, R. Schilling, I. Tyurin (2012)
A Probabilistic Inequality Related to Negative Definite Functions
N. Jacob (1996)
Pseudo-Differential Operators and Markov Processes
Ken-iti Sato (1999)
Lévy Processes and Infinitely Divisible Distributions
Jiange Li, M. Madiman (2016)
A Combinatorial Approach to Small Ball Inequalities for Sums and DifferencesCombinatorics, Probability and Computing, 28
(1964)
Generalized Functions Applications of Harmonic Analysis, vol. 4 Translated by Amiel Feinstein (Academic
Hyunjoong Kim (2017)
Functional Analysis I
R. Strichartz (1994)
A Guide to Distribution Theory and Fourier Transforms
A. Buja, B. Logan, J. Reeds, L. Shepp (1994)
Inequalities and Positive-Definite Functions Arising from a Problem in Multidimensional ScalingAnnals of Statistics, 22
[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
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