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On central limit theorems for Ewens--Pitman model

Yizao Wang

Abstract

We establish a quenched functional central limit theorem for the total number of components of random partitions induced by Chinese restaurant process with parameters $(α,θ), α\in(0,1), θ>-α$. With $P_j$ denoting the asymptotic frequency of $j$-th table, it is well-known that the component count has the same law as the occupancy count of an infinite urn scheme with sampling frequencies being $(P_j)_{j\in\mathbb N}$. Our analysis follows this approach and is based on earlier results of Karlin (1967) and Durieu and Wang (2016). In words, our result reveals that the fluctuations of component count consist of two parts, one due to the sampling effect given the asymptotic frequencies $(P_j)_{j\in\mathbb N}$, the other due to the fluctuations of the random asymptotic frequencies, and in the limit the fluctuations of two parts are conditionally independent given the $α$-diversity. Our result strengthens a recent central limit theorem obtained by Bercu and Favaro (2024) via a different method.

On central limit theorems for Ewens--Pitman model

Abstract

We establish a quenched functional central limit theorem for the total number of components of random partitions induced by Chinese restaurant process with parameters . With denoting the asymptotic frequency of -th table, it is well-known that the component count has the same law as the occupancy count of an infinite urn scheme with sampling frequencies being . Our analysis follows this approach and is based on earlier results of Karlin (1967) and Durieu and Wang (2016). In words, our result reveals that the fluctuations of component count consist of two parts, one due to the sampling effect given the asymptotic frequencies , the other due to the fluctuations of the random asymptotic frequencies, and in the limit the fluctuations of two parts are conditionally independent given the -diversity. Our result strengthens a recent central limit theorem obtained by Bercu and Favaro (2024) via a different method.
Paper Structure (6 sections, 6 theorems, 101 equations)

This paper contains 6 sections, 6 theorems, 101 equations.

Table of Contents

  1. Introduction and main result
  2. Preliminary results on Chinese restaurant process
  3. A quenched functional central limit theorem
  4. Proof for the Poissonized model
  5. De-Poissonization
  6. Joint convergence

Key Result

Theorem 1.1

With the notations above, in $D[0,1]^2$ as $n\to\infty$, where on the right-hand side $Z^{(1)}_\alpha$ and $Z_\alpha^{(2)}$ are two independent Gaussian processes introduced above and independent from $S_\alpha$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof : Proof of Proposition \ref{['prop:CLT Y']} with $\theta>-\alpha,\theta\ne0$
  • Proposition 3.3
  • Lemma 3.4
  • ...and 4 more