Electronic Journal of Probability

Measure concentration through non-Lipschitz observables and functional inequalities

Abstract

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.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 65, 26 pp.

Dates
Accepted: 24 June 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064290

Digital Object Identifier
doi:10.1214/EJP.v18-2425

Mathematical Reviews number (MathSciNet)
MR3078024

Zentralblatt MATH identifier
1282.60024

Rights

Citation

Joulin, Aldéric; Guillin, Arnaud. Measure concentration through non-Lipschitz observables and functional inequalities. Electron. J. Probab. 18 (2013), paper no. 65, 26 pp. doi:10.1214/EJP.v18-2425. https://projecteuclid.org/euclid.ejp/1465064290

References

• Aida, Shigeki; Stroock, Daniel. Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1 (1994), no. 1, 75–86.
• 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.
• Beckner, William. A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 (1989), no. 2, 397–400.
• 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.
• 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.
• 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.
• Bobkov, Sergey; Ledoux, Michel. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998), no. 2, 347–365.
• Bobkov, Sergey; Ledoux, Michel. Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 (2009), no. 2, 403–427.
• Bobkov, Sergey; Madiman, Mokshay. Concentration of the information in data with log-concave distributions. Ann. Probab. 39 (2011), no. 4, 1528–1543.
• Bobkov, Sergey; Tetali, Prasad. Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 (2006), no. 2, 289–336.
• 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.
• Cattiaux, Patrick; Guillin, Arnaud. Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Stat. 12 (2008), 12–29 (electronic).
• 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.
• Cattiaux, Patrick; Guillin, Arnaud; Wu, Liming. Some remarks on weighted logarithmic Sobolev inequality. Indiana Univ. Math. J. 60 (2011), no. 6, 1885–1904.
• 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.
• 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.
• Dai Pra, Paolo; Posta, Gustavo. Entropy decay for interacting systems via the Bochner-Bakry-Ã‰mery 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.
• Diaconis, Persi; Saloff-Coste, Laurent. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996), no. 3, 695–750.
• 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.
• Gross, Leonard. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061–1083.
• 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.
• 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.
• Houdré, Christian. Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 (2002), no. 3, 1223–1237.
• Johnson, Oliver. Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 (2007), no. 6, 791–802.
• Joulin, Aldéric. Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 (2007), no. 3, 782–798.
• Joulin, Aldéric. A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature. Bernoulli 15 (2009), no. 2, 532–549.
• Joulin, Aldéric; Ollivier, Yann. Curvature, concentration and error estimates for Markov chain Monte Carlo. Ann. Probab. 38 (2010), no. 6, 2418–2442.
• Latała, Rafał. Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 (2006), no. 6, 2315–2331.
• Latała, Rafal; Oleszkiewicz, Krzysztof. Between Sobolev and Poincaré. Geometric aspects of functional analysis, 147–168, Lecture Notes in Math., 1745, Springer, Berlin, 2000.
• Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120–216, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
• 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
• Miclo, Laurent. An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999), no. 3, 319–330.
• Ollivier, Yann. Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256 (2009), no. 3, 810–864.
• 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.
• 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).
• Sammer, Marcus D. Aspects of mass transportation in discrete concentration inequalities. Thesis (Ph.D.)â€“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.
• Wu, Liming. Estimate of spectral gap for continuous gas. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 387–409.