On the Markov–krein Identity and Quasi-invariance of the Gamma Process

We present a simple proof of the Markov–Krein identity for distributions of means of linear functionals of the Dirichlet process and its various generalizations. The key idea is to use the representation of the Dirichlet process as the normalized gamma process and fundamental properties of gamma processes. Bibliography: 19 titles. 1. Introduction: Dirichlet processes, gamma processes and the Markov–Krein identity. The purpose of this paper is to apply the presentation of Dirichlet processes as the normalized gamma processes to prove the Markov–Krein identity for distributions of means of Dirichlet processes. This approach turns out to be very fruitful; in particular, it allows us to obtain the desired formula almost immediately. The Dirichlet processes introduced in [4] play a key role in Bayesian nonparametric statistics. The classical definition of these processes is the following. Denote by