This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\infty $ queue. They describe in particular the exponential dissipation of $\Phi $-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/$\infty $ queues. Proofs are elementary and rely essentially on the development of a “$\Phi $-calculus”.

Keywords: functional inequalities, Markov processes, entropy, birth and death processes, queues

@article{PS_2006__10__317_0, author = {Chafa{\"\i}, Djalil}, title = {Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue}, journal = {ESAIM: Probability and Statistics}, pages = {317--339}, publisher = {EDP-Sciences}, volume = {10}, year = {2006}, doi = {10.1051/ps:2006013}, mrnumber = {2247924}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2006013/} }

TY - JOUR AU - Chafaï, Djalil TI - Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue JO - ESAIM: Probability and Statistics PY - 2006 SP - 317 EP - 339 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006013/ DO - 10.1051/ps:2006013 LA - en ID - PS_2006__10__317_0 ER -

Chafaï, Djalil. Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue. ESAIM: Probability and Statistics, Volume 10 (2006), pp. 317-339. doi : 10.1051/ps:2006013. http://www.numdam.org/articles/10.1051/ps:2006013/

[1] On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573-602. | Zbl

and ,[2] Clark-Ocone formulas and Poincaré inequalities on the discrete cube. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 101-137. | Numdam | Zbl

,[3] L'hypercontractivité et son utilisation en théorie des semigroupes. Lectures on probability theory (Saint-Flour, 1992), Lect. Notes Math. 1581 (1994) 1-114. | Zbl

,[4] Moment inequalities for functions of independent random variables. Ann. Probab. 33 (2005) 514-560. | Zbl

, , and ,[5] Spectral gap estimates for interacting particle systems via a Bochner type inequality. J. Funct. Anal. 232 (2006) 222-258. | Zbl

, , and ,[6] On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347-365. | Zbl

and ,[7] Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Ž. 8 (1967) 983-1004. | Zbl

,[8] Modified Log-Sobolev Inequalities in Discrete Settings, Preliminary version appeared in Proc. of the ACM STOC 2003, pp. 287-296. Cf. http://www.math.gatech.edu/~tetali/, 2003.

and ,[9] Markov chains, Gibbs fields, Monte Carlo simulation, and queues. Texts Appl. Math. 31 (1999) xviii+444. | MR | Zbl

,[10] A continuous stochastic maturation model, preprint arXiv math.PR/0412193 or CNRS HAL ccsd-00003498, 2004.

and ,[11] Entropies, convexity, and functional inequalities: on $\Phi $-entropies and $\Phi $-Sobolev inequalities. J. Math. Kyoto Univ. 44 (2004) 325-363. | Zbl

,[12] Variational formulas of Poincaré-type inequalities for birth-death processes. Acta Math. Sin. (Engl. Ser.) 19 (2003) 625-644. | Zbl

,[13] Entropy dissipation estimates in a Zero-Range dynamics, preprint arXiv math.PR/0405455, 2004. | MR | Zbl

and ,[14] Logarithmic Sobolev inequality for zero-range dynamics: independence of the number of particles. Ann. Probab. 33 (2005) 2355-2401. | Zbl

and ,[15] Logarithmic Sobolev inequality for zero-range dynamics. Electron. J. Probab. 10 (2005) 525-576. | Zbl

and ,[16] Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002), 1959-1976. | Zbl

, and ,[17] Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695-750. | Zbl

and ,[18] Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence. | MR | Zbl

and ,[19] Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51-79. | Zbl

,[20] Preservation of log-concavity on summation, preprint arXiv math.PR/0502548, 2005. | MR

and ,[21] On local Poisson-type deviation inequalities for curved continuous time Markov chains, with applications to birth-death processes, personal communication, preprint 2006. | MR

,[22] Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM Probab. Stat. 8 (2004) 87-101 (electronic). | Numdam

and ,[23] Linear growth birth and death processes. J. Math. Mech. 7 (1958) 643-662. | Zbl

and ,[24] Blocking probabilities in large circuit-switched networks. Adv. in Appl. Probab. 18 (1986) 473-505. | Zbl

,[25] Loss networks. Ann. Appl. Probab. 1 (1991) 319-378. | Zbl

,[26] Scaling limits of interacting particle systems. Fundamental Principles of Mathematical Sciences 320, Springer-Verlag, Berlin (1999). | MR | Zbl

and ,[27] Between Sobolev and Poincaré, Geometric aspects of functional analysis. Lect. Notes Math. 1745 (2000) 147-168. | Zbl

and ,[28] Concentration inequalities and model selection, Lectures on probability theory and statistics (Saint-Flour, 2003), available on the author's web-site http://www.math.u-psud.fr/~massart/stf2003_massart.pdf.

,[29] Logarithmic Sobolev inequalities for birth-death process and diffusion process on the line. Chinese J. Appl. Probab. Statist. 18 (2002) 94-100.

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

,[31] Stochastic networks and queues, french ed., Applications of Mathematics (New York) 52, Springer-Verlag, Berlin, 2003, Stochastic Modelling and Applied Probability. | MR | Zbl

,[32] Convex analysis, Princeton Landmarks in Mathematics, Reprint of the 1970 original, Princeton Paperbacks, Princeton University Press (1997) xviii+451. | MR | Zbl

,[33] Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996). Lect. Notes Math. 1665 (1997) 301-413. | Zbl

,[34] A characteristic property of linear growth birth and death processes. The Indian J. Statist. Ser. A 50 (1988) 184-189. | Zbl

,[35] A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000) 427-438. | Zbl

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