[Paper Review] The genealogy of nearly critical branching processes in varying environment

Florin Boenkost, Félix Foutel‐Rodier+1 more|arXiv (Cornell University)|Jul 23, 2022

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes a Yaglom limit law for the rescaled size of a nearly critical branching process in a varying environment, conditional on survival, and proves convergence of the genealogical structure to a time-changed Brownian coalescent point process in the Gromov–Hausdorff–Prohorov topology. The approach uses spinal decomposition and a novel many-to-few formula under second moment conditions, enabling convergence of key genealogical quantities such as population size, time to the most recent common ancestor, and the reduced tree.

ABSTRACT

Building on the spinal decomposition technique in Foutel-Rodier and Schertzer (2022) 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 toprove convergence of the genealogical structure of the population at a fixed time horizon -- when the sequence of trees are envisioned as a sequence of metric spaces -- in the Gromov--Hausdorff--Prohorov (GHP) topology. We characterize the limiting metric space as a time-changed version of the Brownian coalescent point process Popovic (2004). Beyond our specific model, we derive several general results allowing one 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 (Foutel-Rodier and Schertzer 2022), we hope that our specific example illustrates a general methodology that could be applied to more complex branching processes.

Motivation & Objective

• To establish a Yaglom-type limit law for the rescaled population size of a nearly critical branching process in a varying environment, conditional on long-term survival.
• To prove convergence of the genealogical structure—viewed as a random metric measure space—in the Gromov–Hausdorff–Prohorov topology.
• To develop general tools for analyzing random metric spaces in branching processes near criticality, particularly via spinal decomposition and moment methods.
• To unify the convergence of multiple genealogical quantities (e.g., population size, time to MRCA, reduced tree, k-sampled tree) under a single topological framework.
• To demonstrate the robustness of the spinal decomposition method under second moment conditions, extending beyond traditional moment assumptions.

Proposed method

• Utilizes spinal decomposition to reweight the probability measure, enabling analysis of the genealogical structure via the k-spine tree, which captures the joint distribution of k uniformly sampled individuals.
• Applies a many-to-few formula to reduce the k-th moment of the tree to the distribution of a k-spine tree under a change of measure.
• Employs a truncation argument to work under a second moment condition, avoiding the need for all moments to exist.
• Establishes Gromov–Hausdorff–Prohorov convergence by verifying moment conditions and using continuity of mass and metric functionals under the Gromov–weak topology.
• Characterizes the limiting metric space as a time-changed version of the Brownian coalescent point process, derived from a random environment with time-inhomogeneous variance.
• Uses the convergence of the reduced process and the genealogy of k sampled individuals to derive joint convergence of multiple genealogical statistics.

Experimental results

Research questions

• RQ1 Does a Yaglom limit law hold for the rescaled population size of a nearly critical branching process in a varying environment, conditional on survival?
• RQ2 Can the genealogical structure of such a process converge in the Gromov–Hausdorff–Prohorov topology as the population size grows?
• RQ3 How can the spinal decomposition method be adapted to work under second moment conditions rather than full moment assumptions?
• RQ4 What is the limiting structure of the genealogy when the population is conditioned to survive for a long time in a varying environment?
• RQ5 Can the convergence of multiple genealogical quantities—such as the time to the most recent common ancestor and the reduced tree—be derived from a single convergence result in the GHP topology?

Key findings

• The rescaled population size of a nearly critical branching process in a varying environment converges in distribution to an exponential random variable under the Yaglom limit, with mean determined by the integral of the environmental variance over time.
• The genealogical structure converges in the Gromov–Hausdorff–Prohorov topology to a time-changed Brownian coalescent point process, which encodes the ancestral lineages in a continuous-time coalescent framework.
• The number of ancestors at time sN with descendants at time tN converges in distribution to a geometric random variable with success probability depending on the environmental variance over [s,t].
• The sizes of the descendant families of these ancestors converge in distribution to i.i.d. exponential random variables with mean proportional to the integral of the environmental variance over [s,t].
• The convergence of the genealogical metric space implies joint convergence of all key genealogical statistics, including the time to the most recent common ancestor and the reduced tree.
• The method enables the derivation of convergence results for multiple quantities—such as the population size, k-sampled tree, and reduced process—simultaneously from a single limiting object in the GHP topology.