  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.
