Characteristic Functions of Means of Distributions Chosen from a Dirichlet Process

Let P be a Dirichlet process with parameter α on $(R, B)$, where R is the real line, B is the σ-field of Borel subsets of R and α is a non-null finite measure on $(R, B)$. By the use of characteristic functions we show that if Q(·) = α(·)/α(R) is a Cauchy distribution then the mean ∫R x dP(x) has the same Cauchy distribution and that if Q is normal then the distribution of the mean can be roughly approximated by a normal distribution. If the 1st moment of Q exists, then the distribution of the mean is different from Q except for a degenerate case. Similar results hold also in the multivariate case.