The asymptotic distribution of the length of Beta-coalescent trees

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

2011-07-14

arxiv.org/abs/1107.2855v2

Mathematics

Probability

some corrections and additional references, 27 pages

Scientific paper

We derive the asymptotic distribution of the total length $L_n$ of a Beta($2-\alpha,\alpha$)-coalescent tree for $1<\alpha < 2$, starting from $n$ individuals. There are two regimes: If $\alpha \le \tfrac 12 (1+ \sqrt 5)$, 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 non-degenerate limit distribution. As a consequence we obtain the limit distribution of the number $S_n$ of segregation sites.

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