Large Sample Asymptotics for the Two Parameter Poisson Dirichlet Process

This paper explores large sample properties of the two parameter $(\alpha,\theta)$ 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 $\theta+n\alpha.$ This latter result complements large $\theta$ 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.