Functional inequalities for discrete gradients and application to the geometric distribution

Aldéric Joulin; Nicolas Privault

ESAIM: Probability and Statistics (2004)

  • Volume: 8, page 87-101
  • ISSN: 1292-8100

Abstract

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We present several functional inequalities for finite difference gradients, such as a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities, associated deviation estimates, and an exponential integrability property. In the particular case of the geometric distribution on ℕ we use an integration by parts formula to compute the optimal isoperimetric and Poincaré constants, and to obtain an improvement of our general logarithmic Sobolev inequality. By a limiting procedure we recover the corresponding inequalities for the exponential distribution. These results have applications to interacting spin systems under a geometric reference measure.

How to cite

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Joulin, Aldéric, and Privault, Nicolas. "Functional inequalities for discrete gradients and application to the geometric distribution." ESAIM: Probability and Statistics 8 (2004): 87-101. <http://eudml.org/doc/245603>.

@article{Joulin2004,
abstract = {We present several functional inequalities for finite difference gradients, such as a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities, associated deviation estimates, and an exponential integrability property. In the particular case of the geometric distribution on $\{\mathbb \{N\}\}$ we use an integration by parts formula to compute the optimal isoperimetric and Poincaré constants, and to obtain an improvement of our general logarithmic Sobolev inequality. By a limiting procedure we recover the corresponding inequalities for the exponential distribution. These results have applications to interacting spin systems under a geometric reference measure.},
author = {Joulin, Aldéric, Privault, Nicolas},
journal = {ESAIM: Probability and Statistics},
keywords = {geometric distribution; isoperimetry; logarithmic Sobolev inequalities; spectral gap; Herbst method; deviation inequalities; Gibbs measures; Geometric distribution},
language = {eng},
pages = {87-101},
publisher = {EDP-Sciences},
title = {Functional inequalities for discrete gradients and application to the geometric distribution},
url = {http://eudml.org/doc/245603},
volume = {8},
year = {2004},
}

TY - JOUR
AU - Joulin, Aldéric
AU - Privault, Nicolas
TI - Functional inequalities for discrete gradients and application to the geometric distribution
JO - ESAIM: Probability and Statistics
PY - 2004
PB - EDP-Sciences
VL - 8
SP - 87
EP - 101
AB - We present several functional inequalities for finite difference gradients, such as a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities, associated deviation estimates, and an exponential integrability property. In the particular case of the geometric distribution on ${\mathbb {N}}$ we use an integration by parts formula to compute the optimal isoperimetric and Poincaré constants, and to obtain an improvement of our general logarithmic Sobolev inequality. By a limiting procedure we recover the corresponding inequalities for the exponential distribution. These results have applications to interacting spin systems under a geometric reference measure.
LA - eng
KW - geometric distribution; isoperimetry; logarithmic Sobolev inequalities; spectral gap; Herbst method; deviation inequalities; Gibbs measures; Geometric distribution
UR - http://eudml.org/doc/245603
ER -

References

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