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Binomial-Poisson entropic inequalities and the M/M/∞ queue

Published online by Cambridge University Press:  08 September 2006

Djalil Chafaï
Djalil Chafaï*
Affiliation:
UMR 181 INRA/ENVT Physiopathologie et Toxicologie Experimentales, École Nationale Vétérinaire de Toulouse, 23 Chemin des Capelles, 31076, Toulouse Cedex 3, France, and UMR 5583 CNRS/UPS Laboratoire de Statistique et Probabilités, Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, Cedex 4, France. chafai@math.ups-tlse.fr.nospam
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Abstract

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-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/∞ queues.Proofs are elementary and rely essentially on the development of a “Φ-calculus”.

Keywords

Functional inequalitiesMarkov processesentropybirth and death processesqueues.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 317 - 339
Copyright
© EDP Sciences, SMAI, 2006

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References

Ané, C. and Ledoux, M., On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573–602.
Ané, C., Clark-Ocone formulas and Poincaré inequalities on the discrete cube. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 101–137. CrossRef
D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes. Lectures on probability theory (Saint-Flour, 1992), Lect. Notes Math. 1581 (1994) 1–114.
Boucheron, S., Bousquet, O., Lugosi, G. and Massart, P., Moment inequalities for functions of independent random variables. Ann. Probab. 33 (2005) 514–560. CrossRef
Boudou, A.-S., Caputo, P., Dai Pra, P. and Posta, G., Spectral gap estimates for interacting particle systems via a Bochner type inequality. J. Funct. Anal. 232 (2006) 222–258. CrossRef
Bobkov, S.G. and Ledoux, M., On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347–365. CrossRef
Borovkov, A.A., Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Ž. 8 (1967) 983–1004.
S. Bobkov and P. Tetali, 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.
P. Brémaud, Markov chains, Gibbs fields, Monte Carlo simulation, and queues. Texts Appl. Math. 31 (1999) xviii+444.
D. Chafaï and D. Concordet, A continuous stochastic maturation model, preprint arXiv math.PR/0412193 or CNRS HAL ccsd-00003498, 2004.
Chafaï, D., Entropies, convexity, and functional inequalities: on $\Phi$ -entropies and $\Phi$ -Sobolev inequalities. J. Math. Kyoto Univ. 44 (2004) 325–363. CrossRef
M.F. Chen, Variational formulas of Poincaré-type inequalities for birth-death processes. Acta Math. Sin. (Engl. Ser.) 19 (2003) 625–644.
P. Caputo and G. Posta, Entropy dissipation estimates in a Zero-Range dynamics, preprint arXiv math.PR/0405455, 2004.
Dai Pra, P. and Posta, G., Logarithmic Sobolev inequality for zero-range dynamics: independence of the number of particles. Ann. Probab. 33 (2005) 2355–2401. CrossRef
Dai Pra, P. and Posta, G., Logarithmic Sobolev inequality for zero-range dynamics. Electron. J. Probab. 10 (2005) 525–576. CrossRef
P. Dai Pra, A.M. Paganoni and G. Posta, Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002), 1959–1976.
Diaconis, P. and Saloff-Coste, L., Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695–750.
S.N. Ethier and T.G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence.
Goel, S., Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51–79. CrossRef
O. Johnson and C. Goldschmidt, Preservation of log-concavity on summation, preprint arXiv math.PR/0502548, 2005.
A. Joulin, On local Poisson-type deviation inequalities for curved continuous time Markov chains, with applications to birth-death processes, personal communication, preprint 2006.
Joulin, A. and Privault, N., Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM Probab. Stat. 8 (2004) 87–101 (electronic). CrossRef
Karlin, S. and McGregor, J., Linear growth birth and death processes. J. Math. Mech. 7 (1958) 643–662.
Kelly, F.P., Blocking probabilities in large circuit-switched networks. Adv. in Appl. Probab. 18 (1986) 473–505. CrossRef
Kelly, F.P., Loss networks. Ann. Appl. Probab. 1 (1991) 319–378. CrossRef
C. Kipnis and C. Landim, Scaling limits of interacting particle systems. Fundamental Principles of Mathematical Sciences 320, Springer-Verlag, Berlin (1999).
Latała, R. and Oleszkiewicz, K., Between Sobolev and Poincaré, Geometric aspects of functional analysis. Lect. Notes Math. 1745 (2000) 147–168. CrossRef
P. Massart, 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.
Mao, Y., Logarithmic Sobolev inequalities for birth-death process and diffusion process on the line. Chinese J. Appl. Probab. Statist. 18 (2002) 94–100.
Miclo, L., An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319–330.
Ph. Robert, Stochastic networks and queues, french ed., Applications of Mathematics (New York) 52, Springer-Verlag, Berlin, 2003, Stochastic Modelling and Applied Probability.
R.T. Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Reprint of the 1970 original, Princeton Paperbacks, Princeton University Press (1997) xviii+451.
Saloff-Coste, L., Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996). Lect. Notes Math. 1665 (1997) 301–413. CrossRef
Ycart, B., A characteristic property of linear growth birth and death processes. The Indian J. Statist. Ser. A 50 (1988) 184–189.
Wu, L., A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000) 427–438. CrossRef