Theory and numerical analysis for exact distributions of functionals of a Dirichlet process

The distribution of a mean or, more generally, of a vector of means of a Dirichlet process is considered. Some characterizing aspects of this paper are: (i) a review of a few basic results by providing new formulations, free from many of the extra assumptions up to now considered in the literature, and by giving essentially new, simpler and more direct proofs; (ii) new numerical evaluations, with any prescribed error of approximation, of the distribution at issue; (iii) a new form for the law of a vector of means. Moreover, as applications of the results just mentioned, we give: (iv) a sufficient condition to let the distribution of a mean be symmetric; (v) explicit forms for the probability distribution of the variance of the Dirichlet random measure; (vi) some hints to determine the finite-dimensional distributions of a random function connected with Bayesian methods for queuing models.