Large parameter asymptotic analysis for homogeneous normalized random measures with independent increments

Abstract

Homogeneous normalized random measures with independent increments represent a broad class of Bayesian nonparametric priors and thus are widely used. In this article, we obtain the strong law of large numbers, the central limit theorem (CLT), and the functional central limit theorem (fCLT) of such measures when the concentration parameter approaches infinity. To quantify the convergence rate of the CLT we derive, we also study the Berry–Esseen bound, which turns out to be of the form . As an application of the CLT, we present the functional delta method, which can be employed to obtain the limit of the quantile process of these random measures. As an illustration of the CLTs, we demonstrate the convergence numerically for the Dirichlet processes and the normalized inverse Gaussian processes with various choices of the concentration parameters.