The coalescent point process of branching trees

TL;DR

This paper introduces a novel coalescent point process framework for BGW genealogies, linking discrete genealogical structures to continuous-state branching processes and establishing convergence and invariance principles.

Contribution

It defines a new coalescent point process for BGW trees, relates it to spine decompositions, and proves convergence to continuous-state branching process limits.

Findings

The coalescent point process uniquely determines genealogy backwards in time.

The process of point measures converges to a measure associated with CSB processes.

An invariance principle for the coalescent point process is established.

Abstract

We define a doubly infinite, monotone labeling of Bienayme-Galton-Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i; i\ge 1)$, where $A_i$ is the coalescence time between individuals i and i+1. There is a Markov process of point measures $(B_i; i\ge 1)$ 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 $\varepsilon$.