Characterizations of exchangeable partitions and random discrete distributions by deletion properties

Abstract

We prove a long-standing conjecture which characterises the Ewens-Pitman twoparameter 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 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 0r for some sequence of random partitions ( 0r ) 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.