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Scaling limit for small blocks in the Chinese restaurant process

Oleksii Galganov, Andrii Ilienko

TL;DR

This work analyzes the scaling limit of the small-block composition in the Chinese restaurant process by embedding the CRP into random point measures. The authors establish a projective-limit description: the scaled measures converge vaguely to a limiting Poisson framework on an infinite-dimensional cone, with finite-N approximations $\Xi^{(N)}$ guiding the construction and consistency across dimensions. The main result shows the vague convergence of the infinite-dimensional measure $\Xi_n^{(\infty)}$ to $\Xi^{(\infty)}$, enabling both explicit block-count limit theorems (as sums of Poisson variables) and rich functional limit theorems for the dynamic block structure in Skorokhod space. They further dissect singleton behavior and short-lived singletons, obtaining precise distributions and Poisson limits that illuminate the early-time inflationary phase and its impact on long-run CRP dynamics. The framework yields a coherent, tractable description of block composition over time, with explicit probabilistic building blocks (Poisson measures, Poisson random variables, and infinite-dimensional Markov dynamics) that facilitate a broad class of limit theorems for CRP characteristics.

Abstract

The Chinese restaurant process is a basic sequential construction of consistent random partitions. We consider random point measures describing the composition of small blocks in such partitions and show that their scaling limit is given by the projective limit of certain inhomogeneous Poisson measures on cones of increasing dimension. This result makes it possible to derive classical and functional limit theorems in the Skorokhod topology for various characteristics of the Chinese restaurant process.

Scaling limit for small blocks in the Chinese restaurant process

TL;DR

This work analyzes the scaling limit of the small-block composition in the Chinese restaurant process by embedding the CRP into random point measures. The authors establish a projective-limit description: the scaled measures converge vaguely to a limiting Poisson framework on an infinite-dimensional cone, with finite-N approximations guiding the construction and consistency across dimensions. The main result shows the vague convergence of the infinite-dimensional measure to , enabling both explicit block-count limit theorems (as sums of Poisson variables) and rich functional limit theorems for the dynamic block structure in Skorokhod space. They further dissect singleton behavior and short-lived singletons, obtaining precise distributions and Poisson limits that illuminate the early-time inflationary phase and its impact on long-run CRP dynamics. The framework yields a coherent, tractable description of block composition over time, with explicit probabilistic building blocks (Poisson measures, Poisson random variables, and infinite-dimensional Markov dynamics) that facilitate a broad class of limit theorems for CRP characteristics.

Abstract

The Chinese restaurant process is a basic sequential construction of consistent random partitions. We consider random point measures describing the composition of small blocks in such partitions and show that their scaling limit is given by the projective limit of certain inhomogeneous Poisson measures on cones of increasing dimension. This result makes it possible to derive classical and functional limit theorems in the Skorokhod topology for various characteristics of the Chinese restaurant process.

Paper Structure

This paper contains 7 sections, 9 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Proof of Theorem \ref{['TH:MAIN_TH']}
  4. Limit theorems for characteristics of the CRP
  5. Limiting distributions of block counts
  6. Functional limit theorems
  7. Short-lived singletons

Key Result

Theorem 2.1

$\Xi_n^{(N)}$ vaguely converge in distribution as $n\to\infty$ to the Poisson random measure $\Xi^{(N)}$ on $\left(\mathbb{X}_N,\mathcal{B}(\mathbb{X}_N)\right)$ with intensity measure $\mu^{(N)}$.

Figures (3)

  • Figure 1: The representation of $X_1(t)$.
  • Figure 2: The sets $T_{ij}$.
  • Figure 3: A typical sample of atoms of $\Xi^{(1)}$ with the corresponding sample path of $L$.

Theorems & Definitions (19)

  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['TH:MAIN_TH']}
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['prop:counts']}
  • Corollary 4.1
  • ...and 9 more