We consider Kingman's partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by size-biased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the two-parameter family of partition structures.
@cite_1: Invariance of a random discrete distribution under size-biased permutation is equivalent to a conjunction of symmetry conditions on its finite-dimensional distributions. This is applied to characterize residual allocation models with independent factors that are invariant under size-biased permutation. Apart from some exceptional cases and minor modifications, such models form a two-parameter family of generalized Dirichlet distributions.
Given some deletion kernel @math , it is of interest to consider pairs of partition structures @math such that @math reduces @math to @math , meaning that the following extension of formula d-eq holds: where q(n,x):= n ,: ,x d( ,x)p_0( ) ( 1 x n ) is the unconditional probability that the deletion rule removes a part of size @math from @math distributed according to @math . Pitman @cite_1 showed that if @math is a sequence of partition structures such that size-biased deletion reduces @math to @math for each @math , and @math can be represented in terms of random sampling from @math with @math for each @math , then @math is an @math partition as in param2 for some @math and @math , in which case @math is the @math partition structure. This result and Theorem are two different two-parameter generalisations of Kingman's characterisation of @math partition structures. In the result of @cite_1 , the deletion kernel is still defined by size-biased sampling, and repeated deletions generate a succession of partition structures. Whereas in Theorem the deletion kernel is modified, and repeated deletions generate the same partition structure.