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Concentration inequalities for Markov processes via coupling. (English) Zbl 1191.60023

Summary: 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 \(1+\varepsilon\) 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 \(1+\varepsilon\) 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.

MSC:

60E15 Inequalities; stochastic orderings
60J05 Discrete-time Markov processes on general state spaces
60J25 Continuous-time Markov processes on general state spaces
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