Functionals of Dirichlet processes, the Markov Krein Identity and BetaGamma processes
Abstract
This paper describes how one can use the wellknown 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 CauchyStieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, betagamma processes. This represents a generalization of the MarkovKrein identity for mean functionals of Dirichlet processes. A prior to posterior analysis of betagamma processes is given that not only leads to an easy derivation of the MarkovKrein identity, but additionally yields new distributional identities for gamma and betagamma processes. These results give new explanations and intepretations of exisiting results in the literature. This is punctuated by establishing a simple distributional relationship between betagamma and Dirichlet processes.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2003
 arXiv:
 arXiv:math/0310012
 Bibcode:
 2003math.....10012J
 Keywords:

 Mathematics  Probability