Characteristic Functions of Means of Distributions Chosen from a Dirichlet Process

Let P be a Dirichlet process with parameter a on (R, B), where R is the real line, B is the a-field of Borel subsets of R and a is a non-null finite measure on (R, B). By the use of characteristic functions we show that if Q(.) = a( . )/a(R) is a Cauchy distribution then the mean fR 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.