Scaling limit for small blocks in the Chinese restaurant process

TL;DR

This paper investigates the asymptotic behavior of small blocks in the Chinese restaurant process, establishing a scaling limit via inhomogeneous Poisson measures, enabling new limit theorems for the process.

Contribution

It introduces a novel scaling limit for small blocks in the Chinese restaurant process using inhomogeneous Poisson measures, advancing understanding of its asymptotic properties.

Findings

Scaling limit characterized by projective limits of Poisson measures

Derivation of classical limit theorems in the Skorokhod topology

Enhanced understanding of small block composition in the process

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.