MCMC estimation of the law of the mean od a Dirichlet process

The distribution of the mean Gamma_alpha of a Dirichlet process on the real line, with parameter alpha, can be characterized as the invariant distribution of a real Markov chain Gamma_n. In this paper we prove, that, if alpha has finite expectation, the rate of convergence (in total variation) of Gamma_n to Gamma_alpha is geometric. Upper bounds on the rate of convergence are found which seem effective especially in the case where alpha has a support which is not doubly infinite. We use this to study an approximation procedure for the distribution under consideration, and evaluate the approximation error in simulating using this chain. We include examples for a comparison with some of the existing procedures for approximating the distribution considered, and show that the Markov chain approximation compares well with other methods.