The Yule-Kingman Nested Coalescent: Distribution of the Number of Lineages

We consider a model of a population in which individuals are sampled from different species. The Yule-Kingman nested coalescent describes the genealogy of the sample when each species merges with another randomly chosen species with a constant rate $c$ and each pair of lineages belonging to the same species coalesces independently at rate 1. We study the distribution of the number of individual lineages belonging to a given species and show that it converges weakly to a distribution ${\mu }_c^{\ast }$, which is the unique solution to a recursive distributional equation. Furthermore, the average number of individual lineages belonging to each species converges in probability to the mean of ${\mu }_c^{\ast }$.