A Probabilistic Characterization of Negative Definite Functions

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


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).


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

2010 Mathematics Subject Classification

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



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


  1. 1.
    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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    I.M. Gel’fand, N.Ya. Vilenkin, Generalized Functions. Applications of Harmonic Analysis, vol. 4 Translated by Amiel Feinstein (Academic, New York, 1964)CrossRefGoogle Scholar
  3. 3.
    N. Jacob, Pseudo Differential Operators and Markov Processes. Fourier Analysis and Semigroups, vol. I (Imperial College Press, London, 2001)Google Scholar
  4. 4.
    J. Li, M. Madiman, A combinatorial approach to small ball inequalities for sums and differences. Comb. Probab. Comput. 28(1), 100–129 (2019). MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Lifshits, R. Schilling, I. Tyurin, A Probabilistic Inequality Related to Negative Definite Functions. Progress in Probability, vol. 66 (Springer, Basel, 2013), pp. 73–80CrossRefGoogle Scholar
  6. 6.
    W. Rudin, Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. (McGraw-Hill, New York, 1991)Google Scholar
  7. 7.
    K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68 (Cambridge University Press, Cambridge, 1999)Google Scholar
  8. 8.
    I.J. Schoenberg, Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44(3), 522–536 (1938)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R.S. Strichartz, A Guide to Distribution Theory and Fourier Transforms. Reprint of the 1994 original (World Scientific, River Edge, 2003)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IdahoMoscowUSA

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