A New Proof of Convergence of MCMC via the Ergodic Theorem

S. Asmussen and P. W. Glynn

Statistics and Probability Letters, Vol. 81, 1482–1485 (2011)

A key result underlying the theory of MCMC is that any η-irreducible Markov chain having a transition density with respect to η and possessing a stationary distribution is automatically positive Harris recurrent. This paper provides a short self-contained proof of this fact using the ergodic theorem in its standard form as the most advanced tool.