Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits

TL;DR

This paper proves scaling limits for up-down ordered Chinese restaurant processes, extending the model to a three-parameter family and identifying new stationary distributions related to Poisson--Dirichlet partitions.

Contribution

It extends the oCRP model to a three-parameter family and establishes new scaling limit theorems and stationarity results for these processes.

Findings

Limits are self-similar diffusions in a space of interval partitions.

Extended the parameter range for Poisson--Dirichlet distributions to include <.

Connected the processes to applications in Fleming--Viot and tree-valued Markov processes.

Abstract

We establish scaling limit theorems for the up-down ordered Chinese restaurant processes (oCRPs) of Rogers and Winkel as processes in a space of interval partitions. As previously conjectured, the limits are self-similar diffusions previously constructed directly in the continuum. We extend the oCRP model and the results to a three-parameter family ${\rm oCRP}^{(\alpha)}(\theta_1,\theta_2)$, $\alpha\in(0,1)$, $\theta_1,\theta_2\ge 0$. We use the scaling limit approach to extend existing stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $\alpha\in(0,1)$ and $\theta:=\theta_1+\theta_2-\alpha\ge-\alpha$, including for the first time the usual range of $\theta>-\alpha$ rather than being restricted to $\theta\ge 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.