Central limit theorems associated with the hierarchical Dirichlet process

Abstract: The Dirichlet process is a discrete random measure specified by a concentration parameter and a base distribution, and is used as a prior distribution in Bayesian nonparametrics. The hierarchical Dirichlet process generalizes the Dirichlet process by randomizing the base distribution through a draw from another Dirichlet process. It is motivated by the study of groups of clustered data, where the group specific Dirichlet processes are linked through an intergroup Dirichlet process. Focusing on an individual group, the hierarchical Dirichlet process is a discrete random measure whose weights have stronger dependence than the weights of the Dirichlet process. In this paper, we study the asymptotic behavior of the power sum symmetric polynomials for the vector of weights of the hierarchical Dirichlet process when the corresponding concentration parameters tend to infinity. We establish central limit theorems and obtain explicit representations for the asymptotic variances, with the latter clearly showing the impact of the hierarchical structure. These objects are closely related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics.