Weak Consistency of Markov Chain Monte Carlo Methods

We study the asymptotic behavior of non-regular, weak consistent MCMC. It is related to the question whether or not the consistency of MCMC reflects the real behavior of MCMC. We apply weak consistency to a simple mixture model. There is a natural Gibbs sampler which works poorly in simulation. As an alternative, we propose a Metropolis-Hastings (MH) algorithm. We show that the MH algorithm is consistent and the Gibbs sampler is not but weakly consistent. Key fact is that the two MCMC processes tends to a diffusion process and an AR process with respectively. These results come from the weak convergence property of MCMC which is difficult to obtain from Harris recurrence approach.