The Total External Branch Length of Beta-Coalescents

Abstract

For 1 < α < 2 we derive the asymptotic distribution of the total length ofexternalbranches of a Beta(2 − α, α)-coalescent as the numbernof leaves becomes large. It turns out that the fluctuations of the external branch length follow those of τn2−αover the entire parameter regime, where τndenotes the random number of coalescences that bring thenlineages down to one. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at$\alpha_0 = (1+\sqrt 5)/2$([18]).