Abstract

We prove a long-standing conjecture which characterizes 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 πn 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 for some sequence of random partitions which 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 homogeneous neutral-to-the right process.

AMS subject classification (MSC2010) 60C05, 60G09, 05A18

Introduction

Kingman introduced the concept of a partition structure, that is a family of probability distributions for random partitions πn of a positive integer n, with a sampling consistency property as n varies.