## The Annals of Applied Probability

### The asymptotic distribution of the length of Beta-coalescent trees

Götz Kersting

#### Abstract

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

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

Dates
First available in Project Euclid: 12 October 2012

https://projecteuclid.org/euclid.aoap/1350067995

Digital Object Identifier
doi:10.1214/11-AAP827

Mathematical Reviews number (MathSciNet)
MR3025690

Zentralblatt MATH identifier
1251.92034

#### Citation

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. https://projecteuclid.org/euclid.aoap/1350067995

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