A natural derivative on [0,n] and a binomial Poincaré inequality
ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 703-712.

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.

DOI : https://doi.org/10.1051/ps/2014007
Classification : 46N30,  60E15
Mots clés : discrete measures, transportation, poincaré inequalities, Krawtchouk polynomials
@article{PS_2014__18__703_0,
     author = {Hillion, Erwan and Johnson, Oliver and Yu, Yaming},
     title = {A natural derivative on $[0,~n]$ and a binomial Poincar\'e inequality},
     journal = {ESAIM: Probability and Statistics},
     pages = {703--712},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2014007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2014007/}
}
Hillion, Erwan; Johnson, Oliver; Yu, Yaming. A natural derivative on $[0,~n]$ and a binomial Poincaré inequality. ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 703-712. doi : 10.1051/ps/2014007. http://www.numdam.org/articles/10.1051/ps/2014007/

[1] S. Bobkov and F. Götze, Discrete isoperimetric and Poincaré-type inequalities. Probab. Theory Relat. Fields 114 (1999) 245-277. | MR 1701522 | Zbl 0940.60028

[2] S.G. Bobkov, Some extremal properties of the Bernoulli distribution. Teor. Veroyatnost. i Primenen. 41 (1996) 877-884. | MR 1687168 | Zbl 0895.60012

[3] S.G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25 (1997) 206-214. | MR 1428506 | Zbl 0883.60031

[4] S.G. Bobkov and M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347-365. | MR 1636948 | Zbl 0920.60002

[5] T. Cacoullos, On upper and lower bounds for the variance of a function of a random variable. Ann. Probab. 10 (1982) 799-809. | MR 659549 | Zbl 0492.60021

[6] L.H.Y. Chen and J.H. Lou, Characterization of probability distributions by Poincaré-type inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 23 (1987) 91-110. | Numdam | MR 877386 | Zbl 0612.60013

[7] H. Chernoff, A note on an inequality involving the normal distribution. Ann. Probab. 9 (1981) 533-535. | MR 614640 | Zbl 0457.60014

[8] S. Karlin and J. Mcgregor, Ehrenfest urn models. J. Appl. Probab. 2 (1965) 352-376. | MR 184284 | Zbl 0143.40501

[9] C. Klaassen, On an inequality of Chernoff. Ann. Probab. 13 (1985) 966-974. | MR 799431 | Zbl 0576.60015

[10] 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. | MR 1490046 | Zbl 0885.60061

[11] G. Szegő, Orthogonal Polynomials, revised edition. American Mathematical Society, New York (1958). | MR 310533 | Zbl 0023.21505