Asymptotic behavior of clusters in hierarchical species sampling models

Abstract

Consider a random sample of size $N$ from a hierarchical species sampling model. In this paper, we study the large $N$ asymptotic behavior of the number ${\mathbf{K}}_N$ of clusters in the random sample and the number ${\tilde{\mathbf{M}}}_{\ell ,N}$ of clusters represented by exactly $\ell$ latent first-level clusters in the first level of the hierarchical model. In particular, we establish almost sure and $L^p$ convergence for ${\tilde{\mathbf{M}}}_{\ell ,N}$, Gaussian fluctuations and the law of the iterated logarithm for ${\mathbf{K}}_N$, and large deviation principles for both ${\mathbf{K}}_N$ and ${\tilde{\mathbf{M}}}_{\ell ,N}$. Our approach relies on a random sample size (or random index) representation of the number of clusters through the corresponding non-hierarchical species sampling model.

arXiv comment30 pages, comments welcome