Functional central limit theorems for stick-breaking priors

TL;DR

This paper establishes various limit theorems for a range of nonparametric Bayesian priors, including the Dirichlet and Poisson-Dirichlet processes, using moment methods and finite-dimensional distributions.

Contribution

It introduces new conditions for the CLT and FCLT for Dirichlet processes with general weights and extends limit theorems to several complex Bayesian priors.

Findings

Proves strong law of large numbers for these priors.

Establishes CLT and FCLT under new conditions.

Uses moment methods and finite-dimensional distributions for proofs.

Abstract

We obtain the empirical strong law of large numbers, empirical Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various nonparametric Bayesian priors which include the Dirichlet process with general stick-breaking weights, the Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the Dirichlet process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem and functional central limit theorem hold. Except in the case of the generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the method of moments to obtain the convergence results. For the generalized Dirichlet process we use its finite dimensional marginal distributions to obtain the asymptotics although the computations are highly technical.