Asymptotic behavior of clusters in hierarchical species sampling models

TL;DR

This paper investigates the asymptotic properties of the number of clusters and their frequencies in hierarchical species sampling models, providing convergence, fluctuation, and large deviation results.

Contribution

It introduces a novel analysis of cluster behavior in hierarchical models, extending classical species sampling results to the hierarchical setting.

Findings

Almost sure and $L^p$ convergence of cluster counts

Gaussian fluctuation theorems for total clusters

Large deviation principles for cluster counts

Abstract

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