Concentration inequalities for dependent random variables via the martingale method

Concentration inequalities for dependent random variables via the martingale method

Leonid (Aryeh) Kontorovich and Kavita Ramanan

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

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.

Primary Subjects: 60E15

Secondary Subjects: 60J10, 60G42

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

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1229696598
Digital Object Identifier: doi:10.1214/07-AOP384
Mathematical Reviews number (MathSciNet): MR2478678
Zentralblatt MATH identifier: 1154.60310

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