Functional inequalities for discrete gradients and application to the geometric distribution
ESAIM: Probability and Statistics, Volume 8 (2004), pp. 87-101.

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 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.

DOI: 10.1051/ps:2004004
Classification: 60E07, 60E15, 60K35
Keywords: geometric distribution, isoperimetry, logarithmic Sobolev inequalities, spectral gap, Herbst method, deviation inequalities, Gibbs measures
@article{PS_2004__8__87_0,
     author = {Joulin, Ald\'eric and Privault, Nicolas},
     title = {Functional inequalities for discrete gradients and application to the geometric distribution},
     journal = {ESAIM: Probability and Statistics},
     pages = {87--101},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2004},
     doi = {10.1051/ps:2004004},
     mrnumber = {2085608},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2004004/}
}
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
SP  - 87
EP  - 101
VL  - 8
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2004004/
DO  - 10.1051/ps:2004004
LA  - en
ID  - PS_2004__8__87_0
ER  - 
%0 Journal Article
%A Joulin, Aldéric
%A Privault, Nicolas
%T Functional inequalities for discrete gradients and application to the geometric distribution
%J ESAIM: Probability and Statistics
%D 2004
%P 87-101
%V 8
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2004004/
%R 10.1051/ps:2004004
%G en
%F PS_2004__8__87_0
Joulin, Aldéric; Privault, Nicolas. Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM: Probability and Statistics, Volume 8 (2004), pp. 87-101. doi : 10.1051/ps:2004004. http://archive.numdam.org/articles/10.1051/ps:2004004/

[1] S. Bobkov, C. Houdré and P. Tetali, λ , vertex isoperimetry and concentration. Combinatorica 20 (2000) 153-172. | Zbl

[2] S. Bobkov and M. Ledoux, Poincaré's inequalities and Talagrand's concentration phenomenon for the exponential distribution. Probab. Theory Relat. Fields 107 (1997) 383-400. | Zbl

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

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

[5] S.G. Bobkov and C. Houdré, Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997) 184-205. | Zbl

[6] T. Cacoullos and V. Papathanasiou, Characterizations of distributions by generalizations of variance bounds and simple proofs of the CLT. J. Statist. Plann. Inference 63 (1997) 157-171. | Zbl

[7] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N.J. (1970) 195-199. | Zbl

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

[9] P. Dai Pra, A.M. Paganoni and G. Posta, Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002) 1959-1976. | Zbl

[10] P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 (1991) 36-61. | Zbl

[11] P. Fougères, Spectral gap for log-concave probability measures on the real line. Preprint (2002). | MR

[12] L. Gross, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061-1083. | Zbl

[13] C. Houdré, Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 (2002) 1223-1237. | Zbl

[14] C. Houdré and N. Privault, Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli 8 (2002) 697-720. | Zbl

[15] C. Houdré and P. Tetali, Isoperimetric invariants for product Markov chains and graph products. Combinatorica. To appear. | MR | Zbl

[16] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, in Séminaire de Probabilités XXXIII, Lect. Notes Math. 1709 (1999) 120-216. | Numdam | Zbl

[17] L. Miclo, An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | Zbl

[18] T. Stoyanov, Isoperimetric and related constants for graphs and Markov chains. Ph.D. Thesis, Georgia Institute of Technology (2001).

Cited by Sources: