Homogeneous normalized random measures with independent increments (hNRMIs) represent a broad class of Bayesian nonparametric priors and thus are widely used. In this paper, we obtain the strong law of large numbers, the central limit theorem and the functional central limit theorem of hNRMIs when the concentration parameter $a$ approaches infinity. To quantify the convergence rate of the obtained central limit theorem, we further study the Berry-Esseen bound, which turns out to be of the form $O \left( \frac{1}{\sqrt{a}}\right)$. As an application of the central limit theorem, we present the functional delta method, which can be employed to obtain the limit of the quantile process of hNRMIs. As an illustration of the central limit theorems, we demonstrate the convergence numerically for the Dirichlet processes and the normalized inverse Gaussian processes with various choices of the concentration parameters.

翻译：独立增量齐次归一化随机测度（hNRMIs）构成了一类广泛的贝叶斯非参数先验，因此得到广泛应用。本文在浓度参数$a$趋于无穷时，得到了hNRMIs的强大数定律、中心极限定理和泛函中心极限定理。为了量化所得中心极限定理的收敛速度，我们进一步研究了Berry-Esseen界，其形式为$O \left( \frac{1}{\sqrt{a}}\right)$。作为中心极限定理的应用，我们提出了泛函delta方法，可用于推导hNRMIs分位数过程的极限。最后，通过狄利克雷过程和归一化逆高斯过程在不同浓度参数选择下的数值实验，验证了中心极限定理的收敛性。