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.

[1] . 有了样本观察之后, 再计算这个随机概 率测度关于样本观察的条件分布 (后验分布). 当然, 这个先验随机概率测度也不能过于广泛, 我们还 是希望它属于某一类型. 这样, 问题就转化成了如何选择这些先验随机概率测度, 即如何选择非参数 Bayes 模型的先验 (随机) 分布. 关于这点, 很自然地应该遵守以下两个原则:(1) 先验分布的随机概率族空间需要足够大 (使得它能够包含数据所服从的分布);(2) 有了样本观察值之后, 想要得到的后验分布要便于计算和分析. 假定 (Ω, F, P) 是一个概率空间, (X, X ) 是一个可测空间. Dirichlet 过程 {P (A, ω), A ∈ X , ω ∈ Ω} 是一族特殊的随机概率测度: 对几乎所有的 ω ∈ Ω, P (•, ω) 是 (X, X ) 上的概率测度, 而固定 A ∈ X ,

We obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general stickbreaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the stick-breaking 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 marginal distributions to obtain the asymptotics although the computations are highly technical.