• Bernoulli
  • Volume 13, Number 3 (2007), 782-798.

Poisson-type deviation inequalities for curved continuous-time Markov chains

Aldéric Joulin

Full-text: Open access


In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or Γ-curvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birth–death processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below and we extend the deviation inequalities established [Ané, C. and Ledoux, M. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573–602] for continuous-time random walks, seen as models in null curvature. Some applications of these tail estimates are given for Brownian-driven Ornstein–Uhlenbeck processes and M/M/1 queues.

Article information

Bernoulli, Volume 13, Number 3 (2007), 782-798.

First available in Project Euclid: 7 August 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

birth–death process continuous-time Markov chain deviation inequality semigroup Γ-curvature Wasserstein curvature


Joulin, Aldéric. Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 (2007), no. 3, 782--798. doi:10.3150/07-BEJ6039.

Export citation


  • Ané, C. and Ledoux, M. (2000). On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 573–602.
  • Bakry, D. (1997). On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. In New Trends in Stochastic Analysis (Charingworth, 1994), 43–75. River Edge, NJ: World Sci. Publishing.
  • Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. Séminaire de Probabilités XIX 1983/84. Lecture Notes in Math. 1123 177–206. Berlin: Springer.
  • Chafaï, D. (2006). Binomial–Poisson entropic inequalities and the ${M}/{M}/\infty$ queue. ESAIM Probab. Statist. 10 317–339.
  • Chen, M.F. (2004). From Markov Chains to Non-Equilibrium Particle Systems, 2nd edn. River Edge, NJ: World Scientific Publishing Co. Inc.
  • Djellout, H., Guillin, A. and Wu, L. (2004). Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 2702–2732.
  • Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes, Characterization and Convergence. New York: Wiley.
  • Houdré, C. (2002). Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 1223–1237.
  • Houdré, C. and Tetali, P. (2001). Concentration of measure for products of Markov kernels and graph products via functional inequalities. Combin. Probab. Comput. 10 1–28.
  • Kallenberg, O. (1997). Foundations of Modern Probability. New York: Springer.
  • Marton, K. (1996). A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 556–571.
  • Samson, P.M. (2000). Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes. Ann. Probab. 28 416–461.
  • Schmuckenschläger, M. (1998). Martingales, Poincaré type inequalities, and deviation inequalities. J. Funct. Anal. 155 303–323.
  • Sturm, K.T. and Von Renesse, M.K. (2005). Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58 923–940.