## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 4 (1981), 665-670.

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

Robert C. Hannum, Myles Hollander, and Naftali A. Langberg

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 4 (1981), 665-670.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994373

**Digital Object Identifier**

doi:10.1214/aop/1176994373

**Mathematical Reviews number (MathSciNet)**

MR630318

**Zentralblatt MATH identifier**

0473.60019

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K99: None of the above, but in this section

Secondary: 60E05: Distributions: general theory

**Keywords**

Dirichlet process distribution of random functionals

#### Citation

Hannum, Robert C.; Hollander, Myles; Langberg, Naftali A. Distributional Results for Random Functionals of a Dirichlet Process. Ann. Probab. 9 (1981), no. 4, 665--670. doi:10.1214/aop/1176994373. https://projecteuclid.org/euclid.aop/1176994373