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

¹ University of Luxembourg, Campus Kirchberg 1359, Luxembourg
erwan.hillion@uni.lu
² Statistics Group, Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK
o.johnson@bris.ac.uk
³ Department of Statistics, University of California, Irvine, CA 92697, USA
yamingy@uci.edu

Received: 2 July 2013
Revised: 5 January 2014

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.