Convergence rates and moments of Markov chains associated with the mean of Dirichlet processes

We give necessary and sufficient conditions for geometric and polynomial ergodicity of a Markov chain on the real line with invariant distribution Mα equal to the distribution of the mean of a Dirichlet process with parameter α. This extends the applicability of a recent MCMC method for sampling from Mα. We show that the existence of polynomial moments of α is necessary and sufficient for geometric ergodicity, while logarithmic moments of α are necessary and sufficient for polynomial ergodicity. As corollaries it is shown that α and Mα have polynomial moments of the same order, while the order of the logarithmic moments differ by one.