Characteristic functions of means of distributions chosen from a Dirichlet process. (English) Zbl 0535.60012

Yamato, Hajime

Ann. Probab. 12, 262-267 (1984).

Summary: Let P be a Dirichlet process with parameter \(\alpha\) on (R,B), where R is the real line, B is the \(\sigma\)-field of Borel subsets of R and \(\alpha\) is a non-null finite measure on (R,B). By the use of characteristic functions we show that if \(Q(\cdot)=\alpha(\cdot)/\alpha(R)\) is a Cauchy distribution then the mean \(\int_{R}x dP(x)\) 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.

MSC:

60E05	Probability distributions: general theory
60K99	Special processes
60E10	Characteristic functions; other transforms

Keywords:

Dirichlet processes; characteristic functions