The Annals of Applied Probability

The asymptotic distribution of the length of Beta-coalescent trees

Götz Kersting

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We derive the asymptotic distribution of the total length $L_{n}$ of a $\operatorname{Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$, starting from $n$ individuals. There are two regimes: If $\alpha\le\frac{1}{2}(1+\sqrt{5})$, then $L_{n}$ suitably rescaled has a stable limit distribution of index $\alpha$. Otherwise $L_{n}$ just has to be shifted by a constant (depending on $n$) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number $S_{n}$ of segregation sites. These are points (mutations), which are placed on the tree’s branches according to a Poisson point process with constant rate.

Article information

Ann. Appl. Probab., Volume 22, Number 5 (2012), 2086-2107.

First available in Project Euclid: 12 October 2012

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G50: Sums of independent random variables; random walks 60G55: Point processes

Beta-coalescent tree coupling point process stable distribution


Kersting, Götz. The asymptotic distribution of the length of Beta-coalescent trees. Ann. Appl. Probab. 22 (2012), no. 5, 2086--2107. doi:10.1214/11-AAP827.

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