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.
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