The genealogy of a nearly critical branching processes in varying environment

Abstract

Building on the spinal decomposition technique in [10] we prove a Yaglom limit law for the rescaled size of a nearly critical branching process in varying environment conditional on survival. In addition, our spinal approach allows us to prove convergence of the genealogical structure of the population at a fixed time horizon -where the sequence of trees are envisioned as a sequence of metric spaces -in the Gromov-Hausdorff-Prokorov (GHP) topology. We characterize the limiting metric space as a time-changed version of the Brownian coalescent point process [37].Beyond our specific model, we derive several general results allowing to go from spinal decompositions to convergence of random trees in the GHP topology. As a direct application, we show how this type of convergence naturally condenses the limit of several interesting genealogical quantities: the population size, the time to the most-recent common ancestor, the reduced tree and the tree generated by k uniformly sampled individuals. As in a recent article by the authors [10], we hope that our specific example illustrates a general methodology that could be applied to more complex branching processes.