The total external branch length of Beta-coalescents

Abstract

For $1<\alpha <2$ we derive the asymptotic distribution of the total length of {\em external} branches of a Beta$(2-\alpha, \alpha)$-coalescent as the number $n$ of leaves becomes large. It turns out the fluctuations of the external branch length follow those of $\tau_n^{2-\alpha}$ over the entire parameter regime, where $\tau_n$ denotes the random number of coalescences that bring the $n$ lineages down to one. This is in contrast to the fluctuation behavior of the total branch length, which exhibits a transition at $\alpha_0 = (1+\sqrt 5)/2$.