Obtaining Measure Concentration from Markov Contraction

Concentration bounds for non-product, non-Haar measures are fairly recent: the first such result was obtained for contracting Markov chains by Marton in 1996 via the coupling method. The work that followed, with few exceptions, also used coupling. Although this technique is of unquestionable utility as a theoretical tool, it is not always simple to apply. As an alternative to coupling, we use the elementary Markov contraction lemma to obtain simple, useful, and apparently novel concentration results for various Markov-type processes. Our technique consists of expressing probabilities as matrix products and applying Markov contraction to these expressions; thus it is fairly general and holds the potential to yield further results in this vein.

[1]  R. Dobrushin.  Central Limit Theorem for Nonstationary Markov Chains. II , 1956 .

[2]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[3]  Kazuoki Azuma.  WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES , 1967 .

[4]  R. C. Bradley.  Basic Properties of Strong Mixing Conditions , 1985 .

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[7]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[8]  Yuval Peres,et al.  Markov chains indexed by trees , 1994 .

[9]  Tree-Indexed Processes , 1995, math/0404100.

[10]  K. Marton.  Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration , 1996 .

[11]  Steffen L. Lauritzen,et al.  Graphical models in R , 1996 .

[12]  K. Marton.  Measure concentration for a class of random processes , 1998 .

[13]  P. Doukhan,et al.  A new weak dependence condition and applications to moment inequalities , 1999 .

[14]  Paul-Marie Samson,et al.  Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes , 2000 .

[15]  E. Rio.  Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes , 2000 .

[16]  David Gamarnik.  Extension of the PAC framework to finite and countable Markov chains , 2003, IEEE Trans. Inf. Theory.

[17]  C. Külske.  Concentration Inequalities for Functions of Gibbs Fields with Application to Diffraction and Random Gibbs Measures , 2003 .

[18]  K. Marton.  Measure concentration and strong mixing , 2003 .

[19]  P. Kantor.  Foundations of Statistical Natural Language Processing , 2001, Information Retrieval.

[20]  R. C. Bradley.  Basic properties of strong mixing conditions. A survey and some open questions , 2005, math/0511078.

[21]  P. Collet,et al.  Concentration inequalities for random fields via coupling , 2005, math/0503483.

[22]  J. Rosenthal,et al.  On adaptive Markov chain Monte Carlo algorithms , 2005 .

[23]  S. Chatterjee.  Concentration Inequalities With Exchangeable Pairs , 2005 .

[24]  Leonid Kontorovich.  Measure Concentration of Hidden Markov Processes , 2006 .

[25]  Y. Ollivier.  Ricci curvature of Markov chains on metric spaces , 2007, math/0701886.

[26]  Gideon Schechtman,et al.  Measure Concentration of Strongly Mixing Processes with Applications , 2007 .

[27]  Arnaud Guillin,et al.  Transportation-information inequalities for Markov processes , 2007, 0706.4193.

[28]  F. Barthe,et al.  Mass Transport and Variants of the Logarithmic Sobolev Inequality , 2007, 0709.3890.

[29]  J. Chazottes,et al.  Concentration inequalities for Markov processes via coupling , 2008, 0810.0097.

[30]  K. Ramanan,et al.  Concentration Inequalities for Dependent Random Variables via the Martingale Method , 2006, math/0609835.

[31]  A. Kontorovich,et al.  A Strong Law of Large Numbers for Strongly Mixing Processes , 2008, 0807.4665.