  We obtain moment and Gaussian bounds for general 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+epsilon 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+epsilon 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.
