Large sample asymptotics for the two-parameter Poisson–Dirichlet process

Abstract

This paper explores large sample properties of the two-parameter (α, θ) Poisson–Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension of the Dirichlet process, we explore the consistency and weak convergence of the the two-parameter Poisson–Dirichlet posterior process. We also establish the weak convergence of properly centered two-parameter Poisson–Dirichlet processes for large θ+nα. This latter result complements large θ results for the Dirichlet process and Poisson–Dirichlet sequences, and complements a recent result on large deviation principles for the two-parameter Poisson–Dirichlet process. A crucial component of our results is the use of distributional identities that may be useful in other contexts.