  An interesting line of research is the investigation of the laws of random
variables known as Dirichlet means. However, there is not much information on
interrelationships between different Dirichlet means. Here, we introduce two
distributional operations, one of which consists of multiplying a mean
functional by an independent beta random variable, the other being an operation
involving an exponential change of measure. These operations identify
relationships between different means and their densities. This allows one to
use the often considerable analytic work on obtaining results for one Dirichlet
mean to obtain results for an entire family of otherwise seemingly unrelated
Dirichlet means. Additionally, it allows one to obtain explicit densities for
the related class of random variables that have generalized gamma convolution
distributions and the finite-dimensional distribution of their associated
L\'{e}vy processes. The importance of this latter statement is that L\'{e}vy
processes now commonly appear in a variety of applications in probability and
statistics, but there are relatively few cases where the relevant densities
have been described explicitly. We demonstrate how the technique allows one to
obtain the finite-dimensional distribution of several interesting subordinators
which have recently appeared in the literature.
