随机系统树中樱桃和干草叉的渐近联合分布
On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees
摘要：
树形统计提供了有价值的定量洞察进化机制的基础系统发育树，一个常用的图形表示的进化系统，从病毒到物种。通过发展一个版本的扩展Pólya urn模型的极限定理，其中允许负条目的替换矩阵，我们提出了强大数定律和中心极限定理的渐近联合分布的两个子树计数统计量，樱桃的数量和干草叉，随机系统发生树生成的两个广泛使用的零树模型：可区分排列的比例（PDA）和Yule-Harding-Kingman（YHK）模型。我们的研究结果表明，这两个统计量的限制行为，适当缩放时，是独立的初始树中使用的树生成过程。
英文摘要：
Tree shape statistics provide valuable quantitative insights into evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph representation of evolution systems ranging from viruses to species. By developing limit theorems for a version of extended Pólya urn models in which negative entries are permitted for their replacement matrices, we present strong laws of large numbers and central limit theorems for asymptotic joint distributions of two subtree counting statistics, the number of cherries and that of pitchforks, for random phylogenetic trees generated by two widely used null tree models: the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. Our results indicate that the limiting behaviour of these two statistics, when appropriately scaled, are independent of the initial trees used in the tree generating process.
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