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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Buja, B.F. Logan, J.A. Reeds, L.A. Shepp, Inequalities and positive-definite functions arising from a problem in multidimensional scaling. Ann. Stat. 22(1), 406–438 (1994)
I.M. Gel’fand, N.Ya. Vilenkin, Generalized Functions. Applications of Harmonic Analysis, vol. 4 Translated by Amiel Feinstein (Academic, New York, 1964)
N. Jacob, Pseudo Differential Operators and Markov Processes. Fourier Analysis and Semigroups, vol. I (Imperial College Press, London, 2001)
J. Li, M. Madiman, A combinatorial approach to small ball inequalities for sums and differences. Comb. Probab. Comput. 28(1), 100–129 (2019). https://arxiv.org/abs/1601.03927
M. Lifshits, R. Schilling, I. Tyurin, A Probabilistic Inequality Related to Negative Definite Functions. Progress in Probability, vol. 66 (Springer, Basel, 2013), pp. 73–80
W. Rudin, Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. (McGraw-Hill, New York, 1991)
K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68 (Cambridge University Press, Cambridge, 1999)
I.J. Schoenberg, Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44(3), 522–536 (1938)
R.S. Strichartz, A Guide to Distribution Theory and Fourier Transforms. Reprint of the 1994 original (World Scientific, River Edge, 2003)
Acknowledgements
This research was partially supported by a grant from the Simons Foundation, #246211, and an NIH grant P20GM104420.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Gao, F. (2019). A Probabilistic Characterization of Negative Definite Functions. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-26391-1_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-26390-4
Online ISBN: 978-3-030-26391-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)