Lambda-coalescents with dust component

TL;DR

This paper studies lambda-coalescent processes with a dust component, using coupling with Lévy processes and regenerative structures to analyze large-sample properties like absorption time and collisions.

Contribution

It introduces a novel approach coupling lambda-coalescents with Lévy processes to derive large-sample properties, extending understanding of beta-coalescents with a dust component.

Findings

Limit distribution of absorption time derived

Number of collisions characterized

Coupling method with Lévy processes established

Abstract

We consider the lambda-coalescent processes with positive frequency of singleton clusters. The class in focus covers, for instance, the beta$(a,b)$-coalescents with $a>1$. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing L{\'e}vy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.