On Markov Chains with Continuous State Space

In this expository paper, we p r o ve the following theorem, which m a y be of some use in studying Markov chain Monte Carlo methods like hit and run, the Metropolis algorithm, or the Gibbs sampler. Suppose a discrete-time Markov c hain is aperiodic, irreducible, and there is a stationary probability distribution. Then from almost all starting points the distribution of the chain at time n converges in norm to the stationary distribution. This known theorem is a special case of more general results due to Doeblin, and the paper concludes with a brief review of the literature.