A Central Limit Theorem for the Ewens-Pitman random partition in the large-$\theta$ regime via a martingale approach

Abstract

The Ewens-Pitman model defines a distribution on random partitions of $\{1,\dots ,n\}$, with parameters $\alpha \in [0,1)$ and $\theta >-\alpha$; the case $\alpha =0$ reduces to the classical Ewens model from population genetics. We investigate the large-$n$ asymptotic behaviour of the Ewens-Pitman random partition in the nonstandard regime $\theta =\lambda n$ with $\lambda >0$, establishing joint fluctuation results for the total number of blocks $K_n^{\{n\}}$ and the counts $K_{r,n}^{\{n\}}$ of blocks of sizes $r=1,\dots ,d$, for fixed $d\in \mathbb{N}$. In particular, for $\alpha \in [0,1)$ and $\theta =\lambda n$, our main result provides a strong law of large numbers and a central limit theorem for the $(d+1)$-dimensional vector ${\mathbf{K}}_{d,n}^{\{n\}}=(K_n^{\{n\}},K_{1,n}^{\{n\}},\dots ,K_{d,n}^{\{n\}}{)}^T$ as $n\to \infty$. The proof exploits the Chinese restaurant sequential construction under $\theta =\lambda n$ and a central limit theorem for triangular arrays of martingales, extending techniques previously developed for the classical regime with fixed $\theta$. As corollaries of our results, we recover known asymptotics for $K_n^{\{n\}}$ and derive new strong laws and central limit theorems for each fixed $K_{r,n}^{\{n\}}$, thereby completing earlier weak-law results and providing a comprehensive asymptotic description of the Ewens-Pitman partition structure in the large-$\theta$ setting.

28 pages