A ϕ-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replace “almost all” by “all,” which is potentially important when running Markov chain Monte Carlo algorithms. Full-dimensional Metropolis–Hastings algorithms are known to be Harris recurrent. In this paper, we consider conditions under which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are not Harris recurrent. We present a simple but natural two-dimensional counter-example showing how Harris recurrence can fail, and also a variety of positive results which guarantee Harris recurrence. We also present some open problems. We close with a discussion of the practical implications for MCMC algorithms.
"Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains." Ann. Appl. Probab. 16 (4) 2123 - 2139, November 2006. https://doi.org/10.1214/105051606000000510