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

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
Full Text: