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

Erwan Hillion; Oliver Johnson; Yaming Yu

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 703-712
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Hillion, Erwan, Johnson, Oliver, and Yu, Yaming. "A natural derivative on [0, n] and a binomial Poincaré inequality." ESAIM: Probability and Statistics 18 (2014): 703-712. <http://eudml.org/doc/274381>.

@article{Hillion2014,
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.},
author = {Hillion, Erwan, Johnson, Oliver, Yu, Yaming},
journal = {ESAIM: Probability and Statistics},
keywords = {discrete measures; transportation; poincaré inequalities; Krawtchouk polynomials; Poincaré inequalities},
language = {eng},
pages = {703-712},
publisher = {EDP-Sciences},
title = {A natural derivative on [0, n] and a binomial Poincaré inequality},
url = {http://eudml.org/doc/274381},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Hillion, Erwan
AU - Johnson, Oliver
AU - Yu, Yaming
TI - A natural derivative on [0, n] and a binomial Poincaré inequality
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 703
EP - 712
AB - 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.
LA - eng
KW - discrete measures; transportation; poincaré inequalities; Krawtchouk polynomials; Poincaré inequalities
UR - http://eudml.org/doc/274381
ER -

References

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  7. [7] H. Chernoff, A note on an inequality involving the normal distribution. Ann. Probab.9 (1981) 533–535. Zbl0457.60014MR614640
  8. [8] S. Karlin and J. McGregor, Ehrenfest urn models. J. Appl. Probab.2 (1965) 352–376. Zbl0143.40501MR184284
  9. [9] C. Klaassen, On an inequality of Chernoff. Ann. Probab.13 (1985) 966–974. Zbl0576.60015MR799431
  10. [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. Zbl0885.60061MR1490046
  11. [11] G. Szegő, Orthogonal Polynomials, revised edition. American Mathematical Society, New York (1958). Zbl0023.21505MR310533

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