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.
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.
@cite_1
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This article offers a simplified approach to the distribution theory of randomly weighted averages or @math -means @math , for a sequence of i.i.d.random variables @math , and independent random weights @math with @math and @math . The collection of distributions of @math , indexed by distributions of @math , is shown to encode Kingman's partition structure derived from @math . For instance, if @math has Bernoulli @math distribution on @math , the @math th moment of @math is a polynomial function of @math which equals the probability generating function of the number @math of distinct values in a sample of size @math from @math : @math . This elementary identity illustrates a general moment formula for @math -means in terms of the partition structure associated with random samples from @math , first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters @math , found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an @math mean. The analysis of these random means includes the characterization of @math -means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of L 'evy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989).