On the number of segregating sites for populations with large family sizes
On the number of segregating sites for populations with large family sizes
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关于大家庭人口隔离点的数量
DOI：
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发表时间：
2006
影响因子：
1.2
通讯作者：
M. Möhle
中科院分区：
文献类型：
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作者：
M. Möhle
当样本的潜在谱系树是由突变率为0的聚结过程建模时，我们提出了n个个体样本中突变总数sn的递归。幼代被允许同时发生祖先谱系的多次碰撞，这与潜在种群模型中大家族的存在相对应。对于允许多重碰撞的Λ-coalescent过程的子类，使得测度Λ(dx)/x是有限的，我们证明了S n /(nr)在分布上收敛于一个极限变量S，该极限变量S是由某一从属项的指数积分表征的。当测度Λ(dx)/ x2有限时，S的分布与1阶自回归过程的平稳分布一致，并且通过具有特定独立随机系数a和b的形式的随机不动点方程唯一地确定。我们推测，如果测度Λ(dx)/x是无穷大，则概率为S n /E（S n）→1。
We present recursions for the total number, S n , of mutations in a sample of n individuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rate r>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/x is finite, we prove that S n /(nr) converges in distribution to a limiting variable, S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x 2 is finite, the distribution of S coincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the form with specific independent random coefficients A and B. Examples are presented in which explicit representations for (the density of) S are available. We conjecture that S n /E(S n )→1 in probability if the measure Λ(dx)/x is infinite.