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Large sample asymptotics for the two-parameter Poisson--D...
Lancelot F. James · 2007-08-31 · via math.ST updates on arXiv.org

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.