The Total External Branch Length of Beta-Coalescents†

@article{Dahmer2014TheTE,
  title={The Total External Branch Length of Beta-Coalescents†},
  author={Iulia Dahmer and G{\"o}tz Kersting and A. Wakolbinger},
  journal={Combinatorics, Probability and Computing},
  year={2014},
  volume={23},
  pages={1010 - 1027}
}
For 1 < α < 2 we derive the asymptotic distribution of the total length of external branches of a Beta(2 − α, α)-coalescent as the number n of leaves becomes large. It turns out that the fluctuations of the external branch length follow those of τn2−α over the entire parameter regime, where τn denotes the random number of coalescences that bring the n lineages down to one. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at $\alpha_0 = (1… 
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In the Kingman coalescent tree the length of order $r$ is defined as the sum of the lengths of all branches that support $r$ leaves. For $r=1$ these branches are external, while for $r\ge2$ they are…
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