The coalescent structure of uniform and Poisson samples from multitype branching processes

TL;DR

This paper introduces a Poissonization technique to analyze the coalescent structure of samples from multitype branching processes, linking them to multitype $ ext{Lambda}$-coalescents and characterizing their asymptotics.

Contribution

It develops a multitype mixture representation for sampling from branching processes and characterizes the coalescent structure in terms of multitype forests and $ ext{Lambda}$-coalescents.

Findings

Established a Poissonization method for multitype samples.

Characterized the coalescent structure via multitype forests.

Analyzed small time asymptotics linking to $ ext{Lambda}$-coalescents.

Abstract

We introduce a Poissonization method to study the coalescent structure of uniform samples from branching processes. This method relies on the simple observation that a uniform sample of size $k$ taken from a random set with positive Lebesgue measure may be represented as a mixture of Poisson samples with rate $\lambda$ and mixing measure $k \mathrm{d} \lambda/ \lambda$. We develop a multitype analogue of this mixture representation, and use it to characterise the coalescent structure of multitype continuous-state branching processes in terms of random multitype forests. Thereafter we study the small time asymptotics of these random forests, establishing a correspondence between multitype continuous-state branching proesses and multitype $\Lambda$-coalescents.