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A natural derivative on [0, n] and a binomialPoincaré 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. [email protected]
Oliver Johnson
Affiliation:
Statistics Group, Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK; [email protected]
Yaming Yu
Affiliation:
Department of Statistics, University of California, Irvine, CA 92697, USA; [email protected]
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Abstract

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