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A Probabilistic Characterization of Negative Definite Functions

  • Fuchang GaoEmail author
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)

Abstract

It is proved that a continuous function f on \(\mathbb {R}^n\) is negative definite if and only if it is polynomially bounded and satisfies the inequality \(\mathbb {E} f(X-Y)\le \mathbb {E} f(X+Y)\) for all i.i.d. random vectors X and Y  in \(\mathbb {R}^n\). 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).

Keywords

Negative definite function Lévy–Khintchine representation Fourier inversion theorem Polynomially bounded 

2010 Mathematics Subject Classification

Primary: 60E15 42A82; Secondary: 42B10 60E10 

Notes

Acknowledgements

This research was partially supported by a grant from the Simons Foundation, #246211, and an NIH grant P20GM104420.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IdahoMoscowUSA

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