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

Details Distributions of linear functionals of two parameter Poisson--Dirichlet random measures Distributions of linear functionals of two parameter Poisson--Dirichlet random measures

2006-09-18

arxiv.org/abs/math/0609488v4

Annals of Applied Probability 2008, Vol. 18, No. 2, 521-551

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP462 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/07-AAP462

The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson--Dirichlet process $\operatorname {PD}(\alpha,\theta)$. 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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