Shift-coupling and convergence rates of ergodic averages

We study convergence of Markov chains to their stationary distributions . Much recent work has used coupling to get quantitative bounds on the total variation distance between the law and . In this paper, we use shift-coupling to get quantitative bounds on the total variation distance between the the ergodic average law and . This avoids certain problems, related to periodicity and near-periodicity of the Markov chain, which have plagued previous work