Obtaining Measure Concentration from Markov Contraction

Abstract

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.