Distribution of functionals of the the two parameter Poisson-Dirichlet process
Lancelot F. James, Antonio Lijoi, Igor Pruenster · 2006
The present paper provides exact expressions for the probability distribution of linear functionals of the two-parameter Poisson-Dirichlet process PD(�,�). 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 char- acterizations 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 > . . . > 0 and P ∞=1 Pk = 1, denote a sequence of (random) ranked probabilities having the two-parameter (α, θ) Poisson-Dirichlet law, denoted as PD(α, θ) for 0 ≤ α < 1 and θ ≥ 0. A description, as well as a thorough investigation on its properties, is provided in (36). See also (28), (30) and (33). Equivalently, letting Vk, for any k ≥ 1, denote independent random variables such that Vk has beta(1−α, θ+kα) distribution, the PD(α, θ) law is defined as the ranked values of the stick-breaking sequence W1 = V1, Wk = Vk Q k−1 j=1 (1 −Vj) for k ≥ 2. Interestingly PD(α, θ) laws can also be obtained by manipulating random probabilities of the type Pi = Ji/ ˜ T, where ˜ T = P ∞=1 Ji and the sequence (Ji)i≥1 stands for the ranked jumps of a subordinator. If the Ji's are the ranked jumps of a gamma subordinator, then the total mass ˜ T has a gamma distribution with shape θ and scale 1 and (Pi)i≥1 follows a PD(0, θ) law. At the other extreme, letting the Ji's be the ranked jumps of a stable subordinator of index 0 < α < 1, (Pi)i≥1 follows a PD(α,0) distribution. For both α and θ positive, the PD(α, θ) model arises by first taking the ranked jumps governed by the stable subordinator conditioned on their total mass ˜ T and then mixing over a power tempered stable law proportional to t −θ fα(t), where fα(t)