• Bernoulli
  • Volume 16, Number 2 (2010), 361-388.

Dirichlet mean identities and laws of a class of subordinators

Lancelot F. James

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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 Lévy processes. The importance of this latter statement is that Lévy 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.

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Bernoulli, Volume 16, Number 2 (2010), 361-388.

First available in Project Euclid: 25 May 2010

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beta–gamma algebra Dirichlet means and processes exponential tilting generalized gamma convolutions Lévy processes


James, Lancelot F. Dirichlet mean identities and laws of a class of subordinators. Bernoulli 16 (2010), no. 2, 361--388. doi:10.3150/09-BEJ224.

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