Regeneration in Random Combinatorial Structures

TL;DR

This paper advances the theory of Kingman's partition structures by exploring regenerative properties and asymptotic features of ordered and enriched combinatorial models, extending classical results like the Ewens-Pitman family.

Contribution

It introduces the concept of regeneration into Kingman's structures, extending the class of models and analyzing their asymptotic behavior and regenerative properties.

Findings

Regenerative properties of Ewens-Pitman partitions analyzed

Asymptotic features of regenerative compositions studied

Extended models with ordered categories developed

Abstract

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, known as Kingman's paintbox; a central example of the theory - the Ewens-Pitman two-parameter family of 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.