### Distributional Results for Random Functionals of a Dirichlet Process

Robert C. Hannum, Myles Hollander, and Naftali A. Langberg
Source: Ann. Probab. Volume 9, Number 4 (1981), 665-670.

#### Abstract

We obtain an expression for the distribution function of the random variable $\int ZdP$ where $P$ is a random distribution function chosen by Ferguson's (1973) Dirichlet process on $(R, B)$ ($R$ is the real line and $B$ is the $\sigma$-field of Borel sets) with parameter $\alpha$, and $Z$ is a real-valued measurable function defined on $(R, B)$ satisfying $\int |Z| d\alpha < \infty$. As a consequence, we show that when $\alpha$ is symmetric about 0 and $Z$ is an odd function, then the distribution of $\int ZdP$ is symmetric about 0. Our main result is also used to obtain a new result for convergence in distribution of Dirichlet-based random functionals.

First Page:
Primary Subjects: 60K99
Secondary Subjects: 60E05
Full-text: Open access