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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 40 open journal systems 


Concentration inequalities for Markov processes via coupling

Frank Redig, Mathematical Institute Leiden university
Jean Rene Chazottes, CPHT, Ecole Polytechnique, Paris


Abstract
We obtain moment and Gaussian bounds for general coordinate-wise Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order 1+a (a>0) of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order 1+ a is finite, uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.


Full text: PDF

Pages: 1162-1180

Published on: May 31, 2009
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Electronic Journal of Probability. ISSN: 1083-6489