Concentration inequalities for markov processes via coupling

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+å 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+å 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 nonuniform behavior of moments of the coupling time can occur.