Characterizations of exchangeable partitions and random discrete distributions by deletion properties
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 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 πr' for some sequence of random partitions (πr') 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.