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

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

2007-08-31

arxiv.org/abs/0708.4294v2

IMS Collections 2008, Vol. 3, 187-199

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/074921708000000147 the IMS Collections (http://www.imstat.org/publications/imscollec

Scientific paper

10.1214/074921708000000147

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.

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