Measure concentration through non-Lipschitz observables and functional inequalities

Aldéric Joulin (Université de Toulouse)
Arnaud Guillin (Université de Clermont-Ferrand)


Non-Gaussian concentration estimates are obtained for invariant probability measures of reversible Markov processes. We show that the functional inequalities approach combined with a suitable Lyapunov condition allows us to circumvent the classical Lipschitz assumption of the observables. Our method is general and offers an unified treatment of diffusions and pure-jump Markov processes on unbounded spaces.

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Pages: 1-26

Publication Date: June 24, 2013

DOI: 10.1214/EJP.v18-2425


  • Aida, Shigeki; Stroock, Daniel. Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1 (1994), no. 1, 75--86. MR1258492
  • Bakry, Dominique; Barthe, Franck; Cattiaux, Patrick; Guillin, Arnaud. A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case. Electron. Commun. Probab. 13 (2008), 60--66. MR2386063
  • Beckner, William. A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 (1989), no. 2, 397--400. MR0954373
  • Bertini, Lorenzo; Cancrini, Nicoletta; Cesi, Filippo. The spectral gap for a Glauber-type dynamics in a continuous gas. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 1, 91--108. MR1899231
  • Bobkov, Sergey. Spectral gap and concentration for some spherically symmetric probability measures. Geometric aspects of functional analysis, 37--43, Lecture Notes in Math., 1807, Springer, Berlin, 2003. MR2083386
  • Bobkov, Sergey; Ledoux, Michel. Poincaré's inequalities and Talagrand's concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 (1997), no. 3, 383--400. MR1440138
  • Bobkov, Sergey; Ledoux, Michel. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998), no. 2, 347--365. MR1636948
  • Bobkov, Sergey; Ledoux, Michel. Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 (2009), no. 2, 403--427. MR2510011
  • Bobkov, Sergey; Madiman, Mokshay. Concentration of the information in data with log-concave distributions. Ann. Probab. 39 (2011), no. 4, 1528--1543. MR2857249
  • Bobkov, Sergey; Tetali, Prasad. Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 (2006), no. 2, 289--336. MR2283379
  • Caputo, Pietro; Dai Pra, Paolo; Posta, Gustavo. Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 3, 734--753. MR2548501
  • Cattiaux, Patrick; Guillin, Arnaud. Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Stat. 12 (2008), 12--29 (electronic). MR2367991
  • Cattiaux, Patrick; Guillin, Arnaud; Wu, Liming. A note on Talagrand's transportation inequality and logarithmic Sobolev inequality. Probab. Theory Related Fields 148 (2010), no. 1-2, 285--304. MR2653230
  • Cattiaux, Patrick; Guillin, Arnaud; Wu, Liming. Some remarks on weighted logarithmic Sobolev inequality. Indiana Univ. Math. J. 60 (2011), no. 6, 1885--1904. MR3008255
  • Cattiaux, Patrick; Guillin, Arnaud; Zitt, Pierre-André. Poincaré inequalities and hitting times. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), 95-118.
  • Chafaï, Djalil. Entropies, convexity, and functional inequalities: on $\Phi$-entropies and $\Phi$-Sobolev inequalities. J. Math. Kyoto Univ. 44 (2004), no. 2, 325--363. MR2081075
  • Chafaï, Djalil; Joulin, Aldéric. Intertwining and commutation relations for birth-death processes. To appear in Bernoulli.
  • Dai Pra, Paolo; Paganoni, Anna Maria; Posta, Gustavo. Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002), no. 4, 1959--1976. MR1944012
  • Dai Pra, Paolo; Posta, Gustavo. Entropy decay for interacting systems via the Bochner-Bakry-Émery approach. Electron. J. Probab. 18 (2013), 1-21.
  • Decreusefond, Laurent; Joulin, Aldéric; Savy, Nicolas. Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stoch. Anal. 4 (2010), no. 3, 377--399. MR2677197
  • Diaconis, Persi; Saloff-Coste, Laurent. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996), no. 3, 695--750. MR1410112
  • Gao, Fuqing; Guillin, Arnaud; Wu, Liming. Bernstein type's concentration inequalities for symmetric Markov processes. To appear in SIAM Theory Probab. Appl.
  • Gozlan, Nathaël; Léonard, Christian. Transport inequalities. A survey. Markov Process. Related Fields 16 (2010), no. 4, 635--736. MR2895086
  • Gross, Leonard. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061--1083. MR0420249
  • Guillin, Arnaud; Léonard, Christian; Wu, Liming; Yao, Nian. Transportation-information inequalities for Markov processes. Probab. Theory Related Fields 144 (2009), no. 3-4, 669--695. MR2496446
  • Hanson, D. L.; Wright, F. T. A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Statist. 42 1971 1079--1083. MR0279864
  • Houdré, Christian. Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 (2002), no. 3, 1223--1237. MR1920106
  • Johnson, Oliver. Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 (2007), no. 6, 791--802. MR2327839
  • Joulin, Aldéric. Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 (2007), no. 3, 782--798. MR2348750
  • Joulin, Aldéric. A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature. Bernoulli 15 (2009), no. 2, 532--549. MR2543873
  • Joulin, Aldéric; Ollivier, Yann. Curvature, concentration and error estimates for Markov chain Monte Carlo. Ann. Probab. 38 (2010), no. 6, 2418--2442. MR2683634
  • Latała, Rafał. Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 (2006), no. 6, 2315--2331. MR2294983
  • Latała, Rafal; Oleszkiewicz, Krzysztof. Between Sobolev and Poincaré. Geometric aspects of functional analysis, 147--168, Lecture Notes in Math., 1745, Springer, Berlin, 2000. MR1796718
  • Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120--216, Lecture Notes in Math., 1709, Springer, Berlin, 1999. MR1767995
  • Ledoux, Michel. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9 MR1849347
  • Miclo, Laurent. An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999), no. 3, 319--330. MR1710983
  • Ollivier, Yann. Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256 (2009), no. 3, 810--864. MR2484937
  • Otto, Felix; Villani, Cédric. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000), no. 2, 361--400. MR1760620
  • Preston, Chris. Spatial birth-and-death processes. With discussion. Proceedings of the 40th Session of the International Statistical#Institute (Warsaw, 1975), Vol. 2. Invited papers. Bull. Inst. Internat. Statist. 46 (1975), no. 2, 371--391, 405--408 (1975). MR0474532
  • Sammer, Marcus D. Aspects of mass transportation in discrete concentration inequalities. Thesis (Ph.D.)–Georgia Institute of Technology, 2005.
  • Wang, Feng-Yu. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 (1997), no. 3, 417--424. MR1481127
  • Wu, Liming. Estimate of spectral gap for continuous gas. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 387--409. MR2070332

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