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

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 λ and mixing measure $\mathit{k}\mathrm{d}\mathit{\lambda }/\mathit{\lambda }$. 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 processes and multitype Λ-coalescents.