论文标题
不同环境中几乎关键的分支过程的家谱
The genealogy of nearly critical branching processes in varying environment
论文作者
论文摘要
基于Foutel-Rodier和Schertzer（2022）中的脊柱分解技术，我们证明了Yaglom的限制定律，用于重新缩放的大小在不同的环境中以生存条件变化的几乎关键的分支过程。此外，我们的脊柱方法使我们能够在固定时间范围内证明种群的家谱结构的收敛 - 在Gromov -Hausdorff -Prokorov（GHP）拓扑中，树木的序列被视为一系列度量空间。我们将限制度量空间描述为Popovic的Brownian Colescent Point过程的时期版本（2004）。 除了我们的特定模型之外，我们还得出了几个一般结果，可以从脊柱分解到GHP拓扑中随机树的收敛。作为一种直接应用，我们展示了这种类型的收敛如何自然地凝结了几个有趣的族谱量的极限：人口大小，最重心的共同祖先的时间，减少的树木和$ k $均匀抽样的个体产生的树。就像作者最近的一篇文章（Foutel-Rodier and Schertzer 2022）一样，我们希望我们的具体示例说明了可以应用于更复杂的分支过程的一般方法。
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 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 of Popovic (2004). 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 (Foutel-Rodier and Schertzer 2022), we hope that our specific example illustrates a general methodology that could be applied to more complex branching processes.
