Characterizations of exchangeable partitions and random discrete distributions by deletion properties

TL;DR

This paper characterizes the Ewens-Pitman family of exchangeable partitions and related distributions through deletion properties, providing new insights into their structure and regenerative properties.

Contribution

It proves a conjecture that uniquely characterizes the Ewens-Pitman family via a deletion property and explores related Poisson-Dirichlet distributions and regenerative interval partitions.

Findings

Characterization of Ewens-Pitman partitions by deletion property

Identification of independence property for Poisson-Dirichlet distributions

Survey of regenerative properties and interval arrangements

Abstract

We prove a long-standing conjecture which characterises the Ewens-Pitman two-parameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each $n = 2,3, >...$, if one of $n$ individuals is chosen uniformly at random, independently of the random partition $\pi_n$ of these individuals into various types, and all individuals of the same type as the chosen individual are deleted, then for each $r > 0$, given that $r$ individuals remain, these individuals are partitioned according to $\pi_r'$ for some sequence of random partitions $(\pi_r')$ that does not depend on $n$. An analogous result characterizes the associated Poisson-Dirichlet family of random discrete distributions by an independence property related to random deletion of a frequency chosen by a size-biased pick. We also survey the regenerative properties of members of the two-parameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of the Poisson-Dirichlet random sequence into the interval partition induced by the range of a neutral-to-the right process.