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A Martingale Approach to Large-$θ$ Ewens-Pitman Model

Rodrigo Ribeiro

TL;DR

The paper addresses the asymptotic behavior of the number of blocks $K_n$ in the Ewens--Pitman partition under the linear growth regime $\theta=\lambda n$. It develops a martingale-based framework via the Generalized Chinese Restaurant Process to obtain concise LLN and CLT proofs, with refined Berry--Esseen-type bounds. A sharp CLT is established: for $\alpha=0$ the rate is $O(n^{-1/2})$, while for $\alpha\in(0,1)$ the rate is $O(n^{-1/5+\varepsilon})$, addressing a gap for the reinforced regime. The approach yields shorter proofs (compared to prior work) and sharper convergence rates, providing a unified method that highlights the role of martingale structure and conditional variances in large-sample behavior of partition models.

Abstract

We investigate the asymptotic behavior of the number of parts $K_n$ in the Ewens--Pitman partition model under the regime where the diversity parameter is scaled linearly with the sample size, that is, $θ= λn$ for some~$λ> 0$. While recent work has established a law of large numbers (LLN) and a central limit theorem (CLT) for $K_n$ in this regime, we revisit these results through a martingale-based approach. Our method yields significantly shorter proofs, and leads to sharper convergence rates in the CLT, including improved Berry--Esseen bounds in the case $α= 0$, and a new result for the regime $α\in (0,1)$, filling a gap in the literature.

A Martingale Approach to Large-$θ$ Ewens-Pitman Model

TL;DR

The paper addresses the asymptotic behavior of the number of blocks in the Ewens--Pitman partition under the linear growth regime . It develops a martingale-based framework via the Generalized Chinese Restaurant Process to obtain concise LLN and CLT proofs, with refined Berry--Esseen-type bounds. A sharp CLT is established: for the rate is , while for the rate is , addressing a gap for the reinforced regime. The approach yields shorter proofs (compared to prior work) and sharper convergence rates, providing a unified method that highlights the role of martingale structure and conditional variances in large-sample behavior of partition models.

Abstract

We investigate the asymptotic behavior of the number of parts in the Ewens--Pitman partition model under the regime where the diversity parameter is scaled linearly with the sample size, that is, for some~. While recent work has established a law of large numbers (LLN) and a central limit theorem (CLT) for in this regime, we revisit these results through a martingale-based approach. Our method yields significantly shorter proofs, and leads to sharper convergence rates in the CLT, including improved Berry--Esseen bounds in the case , and a new result for the regime , filling a gap in the literature.

Paper Structure

This paper contains 13 sections, 12 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Our Contribution
  4. Discussion on the Rate of Convergence
  5. Proof ideas: A General Principle.
  6. Paper Organization
  7. The Generalized Chinese Restaurant and a useful martingale
  8. Strong Law of Large Numbers for $K_{\lambda n,n}$
  9. Proof of Technical Lemma \ref{['l:sum_phi_asymp']}
  10. Central Limit Theorem for $K_{\lambda n,n}$
  11. Proof of Proposition \ref{['p:beSn']}
  12. Proof of Technical Lemma \ref{['l:sn']}
  13. Classical Theorems

Key Result

Theorem 1

Fix $\lambda > 0$, then for $\alpha \in [0,1)$ almost surely.

Theorems & Definitions (22)

  • Theorem 1: Strong Law of Large Numbers
  • Theorem 2: CLT and Berry-Esseen-type bounds
  • Proposition 1: A useful martingale
  • proof
  • Lemma 1
  • proof : Proof of Theorem \ref{['t:lln']}
  • Lemma 2: Riemann Sum Error Bound
  • proof
  • proof : Proof of Lemma \ref{['l:sum_phi_asymp']}
  • Proposition 2: Berry-Esseen bounds for $S_{n,n}$
  • ...and 12 more