The Dirichlet Process with Large Concentration Parameter

Abstract

Ferguson's Dirichlet process plays an important role in nonparametric Bayesian inference. Let $P_a$ be the Dirichlet process in $\mathbb{R}$ with a base probability measure $H$ and a concentration parameter $a>0.$ In this paper, we show that $\sqrt{a}(P_a((-\infty ,t])-H((-\infty ,t]))$ converges to a certain Brownian bridge as $a\to \infty .$ We also derive a certain Glivenko-Cantelli theorem for the Dirichlet process. Using the functional delta method, the weak convergence of the quantile process is also obtained. A large concentration parameter occurs when a statistician puts too much emphasize on his/her prior guess. This scenario also happens when the sample size is large and the posterior is used to make inference.

16 pages