Dirichlet mean identities and laws of a class of subordinators

TL;DR

This paper introduces new distributional operations that relate different Dirichlet means, enabling explicit density derivations for related random variables and Lévy processes, with applications to subordinators in probability and statistics.

Contribution

It presents novel operations linking Dirichlet means, facilitating the derivation of explicit densities and distributions for related subordinators and Lévy processes.

Findings

Derived explicit densities for generalized gamma convolution distributions

Established relationships between different Dirichlet means

Obtained finite-dimensional distributions of new subordinators

Abstract

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.