Characterizations of exchangeable partitions and random discrete distributions by deletion properties
ALEXANDER V. GNEDIN, Chris Haulk, Jim Pitman · Cambridge University Press eBooks · 2010
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 π 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 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 π n of a positive integer n , with a sampling consistency property as n varies.