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A Gaussian process limit for the self-normalized Ewens-Pitman process

Bernard Bercu, Stefano Favaro

TL;DR

This work analyzes the self-normalized Ewens–Pitman partition process by focusing on the frequencies $P_{r,n}=K_{r,n}/K_n$ and establishing a sharp functional central limit theorem. Building a tailored martingale framework and applying a multidimensional martingale CLT plus the delta-method, the authors show that $\sqrt{K_n}\,((P_{1,n},P_{2,n},\dots)-p_\alpha)$ converges in distribution to a centered Gaussian process $\mathcal{G}(\Gamma_\alpha)$ in $\ell^2$, where $\Gamma_\alpha=\mathrm{diag}(p_\alpha)-p_\alpha p_\alpha^T$ and $p_\alpha$ is the Sibuya distribution. They also derive a practically usable estimator of the stability parameter, $\widehat{\alpha}_n = P_{1,n}$, proving almost-sure consistency and asymptotic normality with variance $\alpha(1-\alpha)$, enabling asymptotic confidence intervals for $\alpha$. The infinite-dimensional analysis hinges on explicit generating functions for the Sibuya law, yielding a closed-form covariance $\Gamma_\alpha$ and enabling the construction of large-$n$ asymptotic uncertainty quantification for the entire self-normalized partition structure. Overall, the paper provides a rigorous Gaussian-process limit for self-normalized Ewens–Pitman frequencies and practical inference tools for the model parameters.

Abstract

For an integer $n\geq1$, consider a random partition $Π_{n}$ of $\{1,\ldots,n\}$ into $K_{n}$ partition sets with $K_{r,n}$ partition subsets of size $r=1,\ldots,n$, and assume $Π_{n}$ distributed according to the Ewens-Pitman model with parameters $α\in]0,1[$ and $θ>-α$. Although the large-$n$ asymptotic behaviors of $K_{n}$ and $K_{r,n}$ are well understood in terms of almost sure convergence and Gaussian fluctuations, much less is known about the asymptotic behavior of $P_{r,n}=K_{r,n}/K_n$ and of the self-normalized Ewens-Pitman process $(P_{1,n},P_{2,n},\dots)$. Motivated by the almost sure convergence of $(P_{1,n},P_{2,n},\dots)$ to the Sibuya distribution $p_α=(p_α(1),p_α(2),\ldots)$, where $p_α(r)$ is the probability mass at $r=1,2,\ldots$, we establish the $\ell^{2}$ distributional convergence \begin{displaymath} \sqrt{K_{n}}((P_{1,n},\,P_{2,n},\ldots)-p_α)\underset{n\rightarrow+\infty}{\overset{\cL}{\longrightarrow}}\mathcal{G}(Γ_α), \end{displaymath} where $\mathcal{G}(Γ_α)$ stands for a centered Gaussian process with covariance matrix $Γ_α=diag(p_α) - p_α p_α^T$. We apply our result to the estimation of the parameter

A Gaussian process limit for the self-normalized Ewens-Pitman process

TL;DR

This work analyzes the self-normalized Ewens–Pitman partition process by focusing on the frequencies and establishing a sharp functional central limit theorem. Building a tailored martingale framework and applying a multidimensional martingale CLT plus the delta-method, the authors show that converges in distribution to a centered Gaussian process in , where and is the Sibuya distribution. They also derive a practically usable estimator of the stability parameter, , proving almost-sure consistency and asymptotic normality with variance , enabling asymptotic confidence intervals for . The infinite-dimensional analysis hinges on explicit generating functions for the Sibuya law, yielding a closed-form covariance and enabling the construction of large- asymptotic uncertainty quantification for the entire self-normalized partition structure. Overall, the paper provides a rigorous Gaussian-process limit for self-normalized Ewens–Pitman frequencies and practical inference tools for the model parameters.

Abstract

For an integer , consider a random partition of into partition sets with partition subsets of size , and assume distributed according to the Ewens-Pitman model with parameters and . Although the large- asymptotic behaviors of and are well understood in terms of almost sure convergence and Gaussian fluctuations, much less is known about the asymptotic behavior of and of the self-normalized Ewens-Pitman process . Motivated by the almost sure convergence of to the Sibuya distribution , where is the probability mass at , we establish the distributional convergence \begin{displaymath} \sqrt{K_{n}}((P_{1,n},\,P_{2,n},\ldots)-p_α)\underset{n\rightarrow+\infty}{\overset{\cL}{\longrightarrow}}\mathcal{G}(Γ_α), \end{displaymath} where stands for a centered Gaussian process with covariance matrix . We apply our result to the estimation of the parameter
Paper Structure (16 sections, 10 theorems, 174 equations)

This paper contains 16 sections, 10 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Estimation of the parameter $\alpha$
  4. Organization of the paper
  5. Proof of Theorem \ref{['main_teo']}
  6. The martingale construction
  7. A central limit theorem
  8. The multi-dimensional martingale construction
  9. A multi-dimensional central limit theorem
  10. A direct application of the delta-method
  11. Determination of the infinite-dimensional covariance matrix
  12. An infinite-dimensional central limit theorem
  13. Conclusion
  14. Sequential or generative construction of the Ewens-Pitman model
  15. Moment formulae for the random variable $(K_{n}, K_{r,n})$
  16. ...and 1 more sections

Key Result

Theorem 1.2

Assume that $\alpha\in]0,1[$ and $\alpha+\theta>0$. Then, we have the $\ell^{2}$ distributional convergence where $\mathcal{G}(\Gamma_\alpha)$ stands for a centered Gaussian process with infinite-dimensional covariance matrix

Theorems & Definitions (19)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 9 more