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

Bernard Bercu, Claudia Contardi, Emanuele Dolera, Stefano Favaro

TL;DR

This paper analyzes the Ewens-Pitman random partition in the nonstandard large-theta regime theta = lambda n with alpha in [0,1). It proves a joint LLN and CLT for the vector comprising the total number of blocks and the counts of blocks of sizes up to d, demonstrating linear scaling in n with explicit limits M_{d,alpha,lambda} and Gaussian fluctuations with covariance Sigma_{d,alpha,lambda}. The approach hinges on a sequential Chinese restaurant process construction in the large-theta setting and a martingale CLT for triangular arrays, adapted from prior fixed-theta techniques. The results recover known asymptotics for K_n^{(n)} and yield new strong laws and CLTs for K_{r,n}^{(n)} (r ≤ d), providing a comprehensive asymptotic description of the Ewens-Pitman partition structure under theta ∝ n. Overall, the work advances understanding of partition structures in nonstandard scaling regimes and offers explicit, computable limits for practical use in probabilistic and combinatorial contexts.

Abstract

The Ewens-Pitman model defines a distribution on random partitions of $\{1,\ldots,n\}$, with parameters $α\in [0,1)$ and $θ> -α$; the case $α=0$ 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 $θ=λn$ with $λ>0$, 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 $α\in[0,1)$ and $θ=λn$, our main result provides a strong law of large numbers and a central limit theorem for the $(d+1)$-dimensional vector $\mathbf{K}_{d,n}^{\{n\}} = \bigl(K_n^{\{n\}}, K_{1,n}^{\{n\}}, \dots, K_{d,n}^{\{n\}}\bigr)^T$ as $n \to \infty$. The proof exploits the Chinese restaurant sequential construction under $θ=λn$ and a central limit theorem for triangular arrays of martingales, extending techniques previously developed for the classical regime with fixed $θ$. 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-$θ$ setting.

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

TL;DR

This paper analyzes the Ewens-Pitman random partition in the nonstandard large-theta regime theta = lambda n with alpha in [0,1). It proves a joint LLN and CLT for the vector comprising the total number of blocks and the counts of blocks of sizes up to d, demonstrating linear scaling in n with explicit limits M_{d,alpha,lambda} and Gaussian fluctuations with covariance Sigma_{d,alpha,lambda}. The approach hinges on a sequential Chinese restaurant process construction in the large-theta setting and a martingale CLT for triangular arrays, adapted from prior fixed-theta techniques. The results recover known asymptotics for K_n^{(n)} and yield new strong laws and CLTs for K_{r,n}^{(n)} (r ≤ d), providing a comprehensive asymptotic description of the Ewens-Pitman partition structure under theta ∝ n. Overall, the work advances understanding of partition structures in nonstandard scaling regimes and offers explicit, computable limits for practical use in probabilistic and combinatorial contexts.

Abstract

The Ewens-Pitman model defines a distribution on random partitions of , with parameters and ; the case reduces to the classical Ewens model from population genetics. We investigate the large- asymptotic behaviour of the Ewens-Pitman random partition in the nonstandard regime with , establishing joint fluctuation results for the total number of blocks and the counts of blocks of sizes , for fixed . In particular, for and , our main result provides a strong law of large numbers and a central limit theorem for the -dimensional vector as . The proof exploits the Chinese restaurant sequential construction under and a central limit theorem for triangular arrays of martingales, extending techniques previously developed for the classical regime with fixed . As corollaries of our results, we recover known asymptotics for and derive new strong laws and central limit theorems for each fixed , thereby completing earlier weak-law results and providing a comprehensive asymptotic description of the Ewens-Pitman partition structure in the large- setting.
Paper Structure (18 sections, 7 theorems, 123 equations)

This paper contains 18 sections, 7 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Related work
  4. Organization of the paper
  5. Proof of Theorem \ref{['main_d']}
  6. Sequential construction of the partition in the large-$\theta$ regime
  7. An important lemma
  8. Proof of the expansion \ref{['mom_m_d']} and of the SLLN \ref{['lln_d']}
  9. Proof of the CLT \ref{['clt_d']}
  10. Construction of the martingale
  11. Completing the proof
  12. The Chinese Restaurant Process
  13. Details on the proof of lemma \ref{['Kxrnn']}
  14. $r \ge 1$, $\alpha \in [0,1)$
  15. $r = 0$, $\alpha \in (0,1)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Fix $d \ge 1$. For $n\in\mathbb{N}$, let $\mathbf{K}_{d,n}^{\{n\}}$ be the random vector containing the number of partition blocks and the number of partition blocks of sizes $1, ..., d$ under the Ewens-Pitman model with parameters $\alpha\in[0,1)$ and $\theta=\lambda n$, with $\lambda>0$. Then, as where $\mathbf{\mathfrak{M}}_{d, \alpha,\lambda} \in \mathbb{R}^{d+1}$and $\Sigma_{d, \alpha, \lamb

Theorems & Definitions (10)

  • Remark
  • Theorem 1.1
  • Corollary 1.2: Theorem 1.2 of Con(25a)
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma A.1
  • proof