The coalescent point process of branching trees

Abstract

We define a doubly infinite, monotone labeling of Bienaymé–Galton–Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_{i};i\ge1)$, where $A_{i}$ is the coalescence time between individuals $i$ and $i+1$. There is a Markov process of point measures $(B_{i};i\ge1)$ keeping track of more ancestral relationships, such that $A_{i}$ is also the first point mass of $B_{i}$. This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in generation $h$ in a planar BGW tree conditioned to survive $h$ generations. The decomposition involves a point measure $\rho$ storing the number of subtrees on the right-hand side of the spine. Under appropriate conditions, we prove convergence of this point measure to a point measure on $\mathbb{R}_{+}$ associated with the limiting continuous-state branching (CSB) process. We prove the associated invariance principle for the coalescent point process, after we discretize the limiting CSB population by considering only points with coalescence times greater than $\varepsilon $. The limiting coalescent point process $(B^{\varepsilon}_{i};i\ge1)$ is the sequence of depths greater than $\varepsilon$ of the excursions of the height process below some fixed level. In the diffusion case, there are no multiple ancestries and (it is known that) the coalescent point process is a Poisson point process with an explicit intensity measure. We prove that in the general case the coalescent process with multiplicities $(B^{\varepsilon}_{i};i\ge1)$ is a Markov chain of point masses and we give an explicit formula for its transition function. The paper ends with two applications in the discrete case. Our results show that the sequence of $A_{i}$’s are i.i.d. when the offspring distribution is linear fractional. Also, the law of Yaglom’s quasi-stationary population size for subcritical BGW processes is disintegrated with respect to the time to most recent common ancestor of the whole population.