Distributions of Functionals of the two Parameter Poisson-Dirichlet Process
Lancelot F. James, Antonio Lijoi, Igor Prünster · RePEc: Research Papers in Economics · 2006
Abstract. The present paper provides exact expressions for the probability distribution of linear functionals of the two–parameter Poisson–Dirichlet process PD(α, θ). Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy– Stieltjes transform are achieved. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean functional of a Poisson–Dirichlet process and the mean functional of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are illustrated. Our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics. 1. Introduction Let (Pi)i≥1, with P1> P2>...