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.
Hajime Yamato. "Characteristic Functions of Means of Distributions Chosen from a Dirichlet Process." Ann. Probab. 12 (1) 262 - 267, February, 1984. https://doi.org/10.1214/aop/1176993389