The asymptotic distribution of the length of Beta-coalescent trees

Abstract

We derive the asymptotic distribution of the total length $L_n$ of a $\mathrm{Beta}(2-\alpha ,\alpha )$-coalescent tree for $1<\alpha <2$, starting from $n$ individuals. There are two regimes: If $\alpha \le 1/2(1+\sqrt{5})$, 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.

Published in at http://dx.doi.org/10.1214/11-AAP827 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Small-time behavior of beta coalescents
• Beta-coalescents and continuous stable random trees

Cited by in corpus (2)

• The fixation line in the $\Lambda$-coalescent
• The Kingman tree length process has infinite quadratic variation