论文标题
一般dirichlet xi-coalescents的频谱的渐近学
Asymptotics of the frequency spectrum for general Dirichlet Xi-coalescents
论文作者
论文摘要
在这项工作中，我们研究了一般的Dirichlet Coalcents，它们是由I.I.D群众分区构建的Xi coalences家族，并且是对称合并的扩展。这类模型是由具有经常性人口瓶颈的人群模型激励的。我们研究了多维块计数过程的短时间行为，该过程的I-The组件计算了大小I的块数量。与标准的合并模型（例如，从无穷大降低的lambda-coalescents类别）相比，我们的过程没有从无穷大的确定性速度。特别是，我们证明，在适当的重新缩放下，它会收敛到随机过程，这是Martingale问题的独特解决方案。我们表明，此限制过程的多元Lamperti变换是Markov添加过程（MAP）。这使我们能够为N点频谱提供一些渐近学，该频谱是广泛用于人群遗传学的统计数据。特别是，重新数量的突变数会收敛于下属的指数功能。
In this work, we study general Dirichlet coalescents, which are a family of Xi-coalecents constructed from i.i.d mass partitions, and are an extension of the symmetric coalescent. This class of models is motivated by population models with recurrent demographic bottlenecks. We study the short time behavior of the multidimensional block counting process whose i-th component counts the number of blocks of size i. Compared to standard coalescent models (such as the class of Lambda-coalescents coming down from infinity), our process has no deterministic speed of coming down from infinity. In particular, we prove that, under appropriate re-scaling, it converges to a stochastic process which is the unique solution of a martingale problem. We show that the multivariate Lamperti transform of this limiting process is a Markov Additive Process (MAP). This allows us to provide some asymptotics for the n-Site Frequency Spectrum, which is a statistic widely used in population genetics. In particular, the rescaled number of mutations converges to the exponential functional of a subordinator.
