  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.
