Obtaining Measure Concentration from Markov Contraction

Details Obtaining Measure Concentration from Markov Contraction Obtaining Measure Concentration from Markov Contraction

2007-11-07

arxiv.org/abs/0711.0987v1

Mathematics

Probability

23 pages

Scientific paper

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. Since then, several other such results have been proved; with few exceptions, these rely on coupling techniques. Though coupling is of unquestionable utility as a theoretical tool, it appears to have some limitations. Coupling has yet to be used to obtain bounds for more general Markov-type processes: hidden (or partially observed) Markov chains, Markov trees, etc. As an alternative to coupling, we apply the elementary Markov contraction lemma to obtain simple, useful, and apparently novel concentration results for the 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 numerous results in this vein.

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