Regenerative Composition Structures

TL;DR

This paper introduces a new class of regenerative composition structures based on a sampling process and regenerative sets, linking their distributions to Lévy processes and exploring their properties and examples.

Contribution

It defines regenerative composition structures via a sampling process and characterizes their distributions in relation to Lévy parameters, including examples from residual models and stochastic processes.

Findings

Characterization of regenerative composition structures

Connection to Lévy processes and regenerative sets

Identification of reversible structures as Bessel bridge excursions

Abstract

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 L{\'e}vy 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 $2 - 2 \alpha$ for some $\alpha \in [0,1]$.