通过脊柱分解研究谱系收敛及其在群体遗传学中的应用
Convergence of genealogies through spinal decomposition with an application to population genetics
AI总结：
本文提出一种基于矩方法和k-spine分解的一般框架，证明分支马尔可夫过程谱系在Gromov-弱拓扑下的收敛性，并将其应用于具有重组的双亲Wright-Fisher模型，揭示其极限行为兼具临界与超临界特征。
摘要：
考虑一个取值于一般类型空间的分支马尔可夫过程。在存活至第 $N$ 代的条件下，现存种群的谱系定义了一个随机标记度量测度空间，其中个体由其类型标记，成对距离由到最近共同祖先的时间度量。在本手稿中，我们设计了一种一般的矩方法来证明当 $N \to \infty$ 时此类谱系在 Gromov-弱拓扑下的收敛性。非正式地说，种群的 $k$ 阶矩是通过在时间 $N$ 以第 $k$ 阶阶乘矩对种群进行大小偏置后，均匀随机选择 $k$ 个个体的谱系来获得的。我们证明所采样的谱系可以用原始分支过程的 $k$-spine 分解来表示，并且收敛性归结为底层 $k$-spines 的收敛性。作为我们框架的一个示例，我们分析了具有重组的双亲 Wright-Fisher 模型的分支近似在大时间尺度下的行为。该模型展现出一些有趣的数学特征。它从超临界状态开始，但自然地趋向临界状态。我们证明其极限行为同时表现出临界和超临界特征。
英文摘要：
Consider a branching Markov process with values in some general type space. Conditional on survival up to generation $N$, the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when $N \to \infty$. Informally, the moment of order $k$ of the population is obtained by observing the genealogy of $k$ individuals chosen uniformly at random after size-biasing the population at time $N$ by its $k$-th factorial moment. We show that the sampled genealogy can be expressed in terms of a $k$-spine decomposition of the original branching process, and that convergence reduces to the convergence of the underlying $k$-spines. As an illustration of our framework, we analyse the large-time behavior of a branching approximation of the biparental Wright-Fisher model with recombination. The model exhibits some interesting mathematical features. It starts in a supercritical state but is naturally driven to criticality. We show that the limiting behavior exhibits both critical and supercritical characteristics.
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