Electronic Communications in Probability

Subgaussian concentration inequalities for geometrically ergodic Markov chains

Abstract

We prove that an irreducible aperiodic Markov chain is geometrically ergodic if and only if any separately bounded functional of the stationary chain satisfies an appropriate subgaussian deviation inequality from its mean.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 64, 12 pp.

Dates
Accepted: 19 September 2015
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465320991

Digital Object Identifier
doi:10.1214/ECP.v20-3966

Mathematical Reviews number (MathSciNet)
MR3407208

Zentralblatt MATH identifier
1329.60251

Rights

Citation

Dedecker, Jérôme; Gouëzel, Sébastien. Subgaussian concentration inequalities for geometrically ergodic Markov chains. Electron. Commun. Probab. 20 (2015), paper no. 64, 12 pp. doi:10.1214/ECP.v20-3966. https://projecteuclid.org/euclid.ecp/1465320991

References

• Adamczak, Radosław. A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 (2008), no. 34, 1000–1034.
• Adamczak, R. and Bednorz, W.: Exponential Concentration Inequalities for Additive Functionals of Markov Chains. (2013). arXiv:1201.3569v2.
• Berkes, Istvan; Philipp, Walter. Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 (1979), no. 1, 29–54.
• Chazottes, Jean-René; GouÃ«zel, Sébastien. Optimal concentration inequalities for dynamical systems. Comm. Math. Phys. 316 (2012), no. 3, 843–889.
• Chazottes, Jean-René; Redig, Frank. Concentration inequalities for Markov processes via coupling. Electron. J. Probab. 14 (2009), no. 40, 1162–1180.
• GouÃ«zel, Sébastien; Melbourne, Ian. Moment bounds and concentration inequalities for slowly mixing dynamical systems. Electron. J. Probab. 19 (2014), no. 93, 30 pp.
• Lezaud, Pascal. Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8 (1998), no. 3, 849–867.
• Lindvall, Torgny. On coupling of discrete renewal processes. Z. Wahrsch. Verw. Gebiete 48 (1979), no. 1, 57–70.
• McDiarmid, Colin. On the method of bounded differences. Surveys in combinatorics, 1989 (Norwich, 1989), 148–188, London Math. Soc. Lecture Note Ser., 141, Cambridge Univ. Press, Cambridge, 1989.
• Merlevède, Florence; Peligrad, Magda; Rio, Emmanuel. A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probab. Theory Related Fields 151 (2011), no. 3-4, 435–474.
• Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6
• Nummelin, E. A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 (1978), no. 4, 309–318.
• Nummelin, Esa. General irreducible Markov chains and nonnegative operators. Cambridge Tracts in Mathematics, 83. Cambridge University Press, Cambridge, 1984. xi+156 pp. ISBN: 0-521-25005-6
• Nummelin, Esa; Tuominen, Pekka. Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stochastic Process. Appl. 12 (1982), no. 2, 187–202.
• Rio, Emmanuel. Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes. (French) [Hoeffding inequalities for Lipschitz functions of dependent sequences] C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 10, 905–908.
• Samson, Paul-Marie. Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes. Ann. Probab. 28 (2000), no. 1, 416–461.