Central limit theorem for the homozygosity of the hierarchical Pitman-Yor process

The hierarchical Pitman-Yor process is a discrete random measure used as a prior in Bayesian nonparametrics. It is motivated by the study of groups of clustered data exhibiting power law behavior. Our focus in this paper is on the Gaussian behavior of a family of statistics, namely the power sum symmetric polynomials for the vector of weights of the process, as the concentration parameters tend to infinity. We establish a central limit theorem and obtain explicit representations for the asymptotic variance, with the latter clearly showing the impact of each component in the hierarchical structure. These results are crucial for understanding the asymptotic behavior of the sampling formulas associated with the process. In comparison with the known results for the hierarchical Dirichlet process, the results for the hierarchical Pitman-Yor process are mathematically more challenging and structurally more revealing of power law behavior.