## The Annals of Probability

### Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes

Paul-Marie Samson

#### Abstract

We prove concentration inequalities for some classes of Markov chains and $\Phi$-mixing processes, with constants independent of the size of the sample, that extend the inequalities for product measures of Talagrand. The method is based on information inequalities put forwardby Marton in case of contracting Markov chains. Using a simple duality argument on entropy, our results also include the family of logarithmic Sobolev inequalities for convex functions. Applications to bounds on supremum of dependent empirical processes complete this work.

#### Article information

Source
Ann. Probab., Volume 28, Number 1 (2000), 416-461.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160125

Digital Object Identifier
doi:10.1214/aop/1019160125

Mathematical Reviews number (MathSciNet)
MR1756011

Zentralblatt MATH identifier
1044.60061

#### Citation

Samson, Paul-Marie. Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes. Ann. Probab. 28 (2000), no. 1, 416--461. doi:10.1214/aop/1019160125. https://projecteuclid.org/euclid.aop/1019160125

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