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

TL;DR

This paper investigates the large sample asymptotic behavior of the two-parameter Poisson--Dirichlet process, focusing on Bayesian posterior consistency, weak convergence, and large deviation principles, with implications for statistical modeling.

Contribution

It provides new large sample asymptotic results for the two-parameter Poisson--Dirichlet process, including posterior convergence and weak convergence under large parameters.

Findings

Posterior process is consistent and weakly convergent.

Established weak convergence for large +nb7.

Complemented existing large deviation results.

Abstract

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.