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Convergence to the Brownian CRT for critical branching Markov processe

Emma Horton, Ellen Powell

TL;DR

This work proves a universal scaling limit for genealogies of continuous-time critical branching Markov processes with finite variance, showing that rescaled genealogies converge to the Brownian continuum random tree under the Gromov-Hausdorff-weak topology. The authors develop a robust martingale framework based on depth-first exploration, spine decompositions, and a functional central limit theorem to connect the discrete tree structure to a Brownian excursion with speed $\sigma^2(f)$. The results extend CRT convergence to a general, continuous-time, non-local branching setting while requiring only second-moment assumptions, and they frame genealogies as metric measure spaces to leverage mm-space convergence theory. The methods offer a flexible, general approach with potential applications to population genetics and stochastic process modeling where universal scaling limits of genealogies are of interest.

Abstract

We prove an invariance principle for a general class of continuous time critical branching processes with finite variance (non-local) branching mechanism. We show that the genealogical trees, viewed as random compact metric measure spaces, converge under rescaling to the Brownian continuum random tree in the Gromov-Hausdorff-weak topology, establishing a universal scaling limit for critical finite variance branching processes.

Convergence to the Brownian CRT for critical branching Markov processe

TL;DR

This work proves a universal scaling limit for genealogies of continuous-time critical branching Markov processes with finite variance, showing that rescaled genealogies converge to the Brownian continuum random tree under the Gromov-Hausdorff-weak topology. The authors develop a robust martingale framework based on depth-first exploration, spine decompositions, and a functional central limit theorem to connect the discrete tree structure to a Brownian excursion with speed . The results extend CRT convergence to a general, continuous-time, non-local branching setting while requiring only second-moment assumptions, and they frame genealogies as metric measure spaces to leverage mm-space convergence theory. The methods offer a flexible, general approach with potential applications to population genetics and stochastic process modeling where universal scaling limits of genealogies are of interest.

Abstract

We prove an invariance principle for a general class of continuous time critical branching processes with finite variance (non-local) branching mechanism. We show that the genealogical trees, viewed as random compact metric measure spaces, converge under rescaling to the Brownian continuum random tree in the Gromov-Hausdorff-weak topology, establishing a universal scaling limit for critical finite variance branching processes.
Paper Structure (17 sections, 25 theorems, 147 equations, 1 figure)

This paper contains 17 sections, 25 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Branching Markov Processes
  3. Main result
  4. Literature
  5. Outline of the paper.
  6. Preliminaries and existing results
  7. Spine decomposition
  8. Limit theorems for critical MBPs
  9. A functional central limit theorem
  10. Carré du Champ operator
  11. Labelling
  12. Depth-first exploration.
  13. Metric measure spaces and convergence
  14. Convergence to the CRT
  15. The Martingale
  16. ...and 2 more sections

Key Result

Theorem 1

Under Assumption A:main we have with respect to the Gromov-Hausdorff-weak topology.

Figures (1)

  • Figure :

Theorems & Definitions (40)

  • Theorem 1
  • Lemma 2: Many-to-one formula
  • Proposition 3: Spine decomposition
  • Lemma 4
  • Theorem 5: Kolmogorov survival probability
  • Theorem 6: Yaglom limit
  • Corollary 7
  • Proposition 8
  • Theorem 9
  • Theorem 10: Functional Central Limit Theorem
  • ...and 30 more