Regenerative partition structures

Abstract

A partition structure is a sequence of probability distributions for π n, a random partition of n, such that if π n is regarded as a random allocation of n unlabeled balls into some random number of unlabeled boxes, and given π n some x of the n balls are removed by uniform random deletion without replacement, the remaining random partition of n - x is distributed like π n-x, for all 1 ≤ x ≤ n. We call a partition structure regenerative if for each n it is possible to delete a single box of balls from π n in such a way that for each 1 ≤ x ≤ n, given the deleted box contains x balls, the remaining partition of n - x balls is distributed like π n-x. Examples are provided by the Ewens partition structures, which Kingman characterised by regeneration with respect to deletion of the box containing a uniformly selected random ball. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) is associated in turn with a regenerative random subset of the positive halfline. Such a regenerative random set is the closure of the range of a subordinator (that is an increasing process with stationary independent increments). The probability distribution of a general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator, for which exponent an integral representation is provided by the Lévy-Khintchine formula. The extended Ewens family of partition structures, previously studied by Pitman and Yor, with two parameters (α, θ), is characterised for 0 ≤ α < 1 and θ > 0 by regeneration with respect to deletion of each distinct part of size x with probability proportional to (n - x)τ + x(1 - τ), where τ α/(α + θ).