  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.
