For each finite measure on [0

A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0, 1] generated by excursions of a standard Bessel bridge of dimension 2 − 2α for some α ∈ [0, 1].

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.

Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursion lengths prior to T is the same as if T were a fixed time. It follows that the generalized arc-sine laws of Lamperti extend to such random times T . For some other random times T , absolute continuity relations are obtained which relate the law of the relative lengths at time T to the law at a fixed time. 1 Introduction Following Lamperti [10], Wendel [24], Kingman [7], Knight [8], PermanPitman -Yor [12, 13, 15], consider the sequence V 1 (T ) V 2 (T ) \Delta \Delta \Delta (1) of ranked lengths of component intervals of the set [0; T ]nZ, where T is a strictly positive random time, and Z is the zero set of a Markov process X started at zero, such as a Brownian motion or Bessel process,...

Abstract The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ∩[0, 1] for a self-similar random set S ⊂ R+ are those which are consistent with respect to a simple truncation operation. Using the standard coding of compositions by finite strings of binary digits starting with a 1, the random composition of n is defined by the first n terms of a random binary sequence of infinite length. The locations of 1s in the sequence are the places visited by an increasing time-homogeneous Markov chain on the positive integers if and only if S = exp(−W) for some stationary regenerative random subset W of the real line. Complementing our study in previous papers, we identify self-similar Markovian composition structures associated with the two-parameter family of partition structures. 1

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From random sets to continuous tensor products: answers to three questions of W. Arveson

Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the Ewens-Pitman two-parameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of categories, • extending the class of tractable models by exploring the idea of regeneration, • analysing regenerative properties of the Ewens-Pitman partitions, • studying asymptotic features of the regenerative compositions. 1