  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.
