I show how to run an N-time-step Markov chain simulation in a circular fashion, so that the state at time 0 follows the state at time N ?1 in the same way as states at times t follow those at times t?1 for 0 < t < N. This wrap-around of the chain is achieved using a coupling procedure, and produces states that all have close to the equilibrium distribution of the Markov chain, under the assumpt...