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April 2005 Functionals of Dirichlet processes, the Cifarelli–Regazzini identity and Beta-Gamma processes
Lancelot F. James
Ann. Statist. 33(2): 647-660 (April 2005). DOI: 10.1214/009053604000001237


Suppose that Pθ(g) is a linear functional of a Dirichlet process with shape θH, where θ>0 is the total mass and H is a fixed probability measure. This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and a posterior calculus for Gamma processes to ascertain properties of linear functionals of Dirichlet processes. In particular, in conjunction with a Gamma identity, we show easily that a generalized Cauchy–Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, Beta-Gamma processes. This represents a generalization of an identity due to Cifarelli and Regazzini, which is also known as the Markov–Krein identity for mean functionals of Dirichlet processes. These results also provide new explanations and interpretations of results in the literature. The identities are analogues to quite useful identities for Beta and Gamma random variables. We give a result which can be used to ascertain specifications on H such that the Dirichlet functional is Beta distributed. This avoids the need for an inversion formula for these cases and points to the special nature of the Dirichlet process, and indeed the functional Beta-Gamma calculus developed in this paper.


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Lancelot F. James. "Functionals of Dirichlet processes, the Cifarelli–Regazzini identity and Beta-Gamma processes." Ann. Statist. 33 (2) 647 - 660, April 2005.


Published: April 2005
First available in Project Euclid: 26 May 2005

zbMATH: 1071.62026
MathSciNet: MR2163155
Digital Object Identifier: 10.1214/009053604000001237

Primary: 62G05
Secondary: 62F15

Keywords: Beta-Gamma processes , Dirichlet process , gamma process , Markov–Krein identity

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 2 • April 2005
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