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A natural derivative on [0, n] and a binomial Poincaré inequality

Published online by Cambridge University Press:  22 October 2014

Erwan Hillion ,
Oliver Johnson  and
Yaming Yu
Erwan Hillion
Affiliation:
University of Luxembourg, Campus Kirchberg 1359, Luxembourg. erwan.hillion@uni.lu
Oliver Johnson
Affiliation:
Statistics Group, Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK; o.johnson@bris.ac.uk
Yaming Yu
Affiliation:
Department of Statistics, University of California, Irvine, CA 92697, USA; yamingy@uci.edu
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Abstract

We consider probability measures supported on a finite discrete interval [0, n]. We introduce a new finite difference operator ∇n, defined as a linear combination of left and right finite differences. We show that this operator ∇n plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures.

Keywords

Discrete measurestransportationPoincaré inequalitiesKrawtchouk polynomials
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 703 - 712
Copyright
© EDP Sciences, SMAI 2014

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References

Bobkov, S. and Götze, F., Discrete isoperimetric and Poincaré-type inequalities. Probab. Theory Relat. Fields 114 (1999) 245–277. Google Scholar
Bobkov, S.G., Some extremal properties of the Bernoulli distribution. Teor. Veroyatnost. i Primenen. 41 (1996) 877–884. Google Scholar
Bobkov, S.G., An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25 (1997) 206–214. Google Scholar
Bobkov, S.G. and Ledoux, M., On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347–365. Google Scholar
Cacoullos, T., On upper and lower bounds for the variance of a function of a random variable. Ann. Probab. 10 (1982) 799–809. Google Scholar
Chen, L.H.Y. and Lou, J.H., Characterization of probability distributions by Poincaré-type inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 23 (1987) 91–110. Google Scholar
Chernoff, H., A note on an inequality involving the normal distribution. Ann. Probab. 9 (1981) 533–535. Google Scholar
Karlin, S. and McGregor, J., Ehrenfest urn models. J. Appl. Probab. 2 (1965) 352–376. Google Scholar
Klaassen, C., On an inequality of Chernoff. Ann. Probab. 13 (1985) 966–974. Google Scholar
L. Saloff-Coste, Lectures on finite Markov Chains, in Lect. Probab. Theory Stat., edited by P. Bernard, St-Flour 1996, in Lect. Notes Math. Springer Verlag (1997) 301–413.
G. Szegő, Orthogonal Polynomials, revised edition. American Mathematical Society, New York (1958).