Functionals of Dirichlet processes, the Markov Krein Identity and Beta-Gamma processes

Abstract

This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and less known results for the gamma process, to address the formidable problem of assessing the distribution 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 the Markov-Krein identity for mean functionals of Dirichlet processes. A prior to posterior analysis of beta-gamma processes is given that not only leads to an easy derivation of the Markov-Krein identity, but additionally yields new distributional identities for gamma and beta-gamma processes. These results give new explanations and intepretations of exisiting results in the literature. This is punctuated by establishing a simple distributional relationship between beta-gamma and Dirichlet processes.