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The Dirichlet Process with Large Concentration Parameter

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 $\sqrt {a} \big(P_a((-\infty,t]) -H((-\infty,t])\big)$ 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.