  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.
