The Annals of Probability

Concentration inequalities for dependent random variables via the martingale method

Leonid (Aryeh) Kontorovich and Kavita Ramanan

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Abstract

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the way, bounds are obtained on martingale differences associated with the random sequences, which may be of independent interest. As applications of the main result, concentration inequalities are also derived for inhomogeneous Markov chains and hidden Markov chains, and an extremal property associated with their martingale difference bounds is established. This work complements and generalizes certain concentration inequalities obtained by Marton and Samson, while also providing different proofs of some known results.

Article information

Source
Ann. Probab., Volume 36, Number 6 (2008), 2126-2158.

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1229696598

Digital Object Identifier
doi:10.1214/07-AOP384

Mathematical Reviews number (MathSciNet)
MR2478678

Zentralblatt MATH identifier
1154.60310

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G42: Martingales with discrete parameter

Keywords
Concentration inequality McDiarmid’s bound bounded martingale differences Markov chains contracting Markov chains hidden Markov chains mixing coefficients

Citation

Kontorovich, Leonid (Aryeh); Ramanan, Kavita. Concentration inequalities for dependent random variables via the martingale method. Ann. Probab. 36 (2008), no. 6, 2126--2158. doi:10.1214/07-AOP384. https://projecteuclid.org/euclid.aop/1229696598


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