Distributional Results for Random Functionals of a Dirichlet Process

We obtain an expression for the distribution function of the random variable ∫ 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 σ-field of Borel sets) with parameter α, and Z is a real-valued measurable function defined on $(R, B)$ satisfying ∫ |Z| dα < ∞. As a consequence, we show that when α is symmetric about 0 and Z is an odd function, then the distribution of ∫ 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.