论文标题
Yule树中外部分支长度的分布
Distribution of external branch lengths in Yule trees
论文作者
论文摘要
Yule分支过程是人口遗传学中随机产生基因树拓扑结构的经典模型。它生成了二进制排名的树（也称为“历史”），其中有有限的叶子叶子。我们研究长度$ \ ell_1> \ ell_2> ...> \ ell_k> ... ... yule生成的大小$ n $的随机历史记录的外部分支的$，其中外部分支的长度定义为其父节点的等级。当$ n \ rightArrow \ infty $时，我们表明，随机变量$ \ ell_k $，一旦将$ \ frac {n- \ ell_k} {\ sqrt {\ sqrt {n/2}} $重新定制，遵循$ 2K $ nim of y Mathbb e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e el y n viral n friafe and n viral n。 $ \ mathbb v（\ ell_k）\ sim n \ big（k- \ frac {πk^2} {16^k} \ binom {2k} {2k} {k}^2 \ big）$。我们的结果有助于研究Yule产生的基因树的组合特征，其中外部分支与影响各个基因拷贝的单胎突变有关。
The Yule branching process is a classical model for the random generation of gene tree topologies in population genetics. It generates binary ranked trees -- also called "histories" -- with a finite number $n$ of leaves. We study the lengths $\ell_1 > \ell_2 > ... > \ell_k > ...$ of the external branches of a Yule generated random history of size $n$, where the length of an external branch is defined as the rank of its parent node. When $n \rightarrow \infty$, we show that the random variable $\ell_k$, once rescaled as $\frac{n-\ell_k}{\sqrt{n/2}}$, follows a $χ$-distribution with $2k$ degrees of freedom, with mean $\mathbb E(\ell_k) \sim n$ and variance $\mathbb V(\ell_k) \sim n \big(k-\frac{πk^2}{16^k} \binom{2k}{k}^2\big)$. Our results contribute to the study of the combinatorial features of Yule generated gene trees, in which external branches are associated with singleton mutations affecting individual gene copies.
