The Annals of Applied Probability

Distributions of linear functionals of two parameter Poisson–Dirichlet random measures

Lancelot F. James, Antonio Lijoi, and Igor Prünster

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The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson–Dirichlet process PD(α, θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson–Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in the paper is demonstrated by means of some illustrative examples. Indeed, 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.

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Ann. Appl. Probab., Volume 18, Number 2 (2008), 521-551.

First available in Project Euclid: 20 March 2008

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Zentralblatt MATH identifier

Primary: 60G57: Random measures
Secondary: 62F15: Bayesian inference 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; Lévy processes

α-stable subordinator Bayesian nonparametric statistics Cauchy–Stieltjes transform Cifarelli–Regazzini identity functionals of random probability measures occupation times Poisson–Dirichlet process


James, Lancelot F.; Lijoi, Antonio; Prünster, Igor. Distributions of linear functionals of two parameter Poisson–Dirichlet random measures. Ann. Appl. Probab. 18 (2008), no. 2, 521--551. doi:10.1214/07-AAP462.

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