Asymptotic behavior of clusters in hierarchical species sampling models

Consider a sample of size $N$ from a population governed by a hierarchical species sampling model. We study the large $N$ asymptotic behavior of the number ${\mathbf{K}}_N$ of clusters and the number ${\mathbf{M}}_{r,N}$ of clusters with frequency $r$ in the sample. In particular, we show almost sure and $L^p$ convergence for ${\mathbf{M}}_{r,N}$, obtain Gaussian fluctuation theorems for ${\mathbf{K}}_N$, and establish large deviation principles for both ${\mathbf{K}}_N$ and ${\mathbf{M}}_{r,N}$. Our approach relies on a random sample size representation of the number of clusters through the corresponding non-hierarchical species sampling model.