Mathematics – Probability
Scientific paper
2007-02-14
Annals of Applied Probability 2006, Vol. 16, No. 4, 2123-2139
Mathematics
Probability
Published at http://dx.doi.org/10.1214/105051606000000510 in the Annals of Applied Probability (http://www.imstat.org/aap/) by
Scientific paper
10.1214/105051606000000510
A $\phi$-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.
Roberts Gareth O.
Rosenthal Jeffrey S.
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