The genealogy of nearly critical branching processes in varying environment
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914062853472256 |
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| author | Boenkost, Florin Foutel-Rodier, Félix Schertzer, Emmanuel |
| author_facet | Boenkost, Florin Foutel-Rodier, Félix Schertzer, Emmanuel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_11612 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The genealogy of nearly critical branching processes in varying environment Boenkost, Florin Foutel-Rodier, Félix Schertzer, Emmanuel Probability Primary 60J80, Secondary 60J90, 60F17 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. |
| title | The genealogy of nearly critical branching processes in varying environment |
| topic | Probability Primary 60J80, Secondary 60J90, 60F17 |
| url | https://arxiv.org/abs/2207.11612 |