Large sample asymptotics for the two-parameter Poisson–Dirichlet process
Lancelot F. James
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.
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Primary Subjects: 62G05
Secondary Subjects: 62F15
Keywords: Bayesian consistency; multiplier CLT; two-parameter Poisson–Dirichlet process; weak convergence
Full-text: Open access
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Permanent link to this document: http://projecteuclid.org/euclid.imsc/1209398469
Digital Object Identifier: doi:10.1214/074921708000000147
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