The Total External Branch Length of Beta-Coalescents

Abstract

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 τ2−α n 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 α0 = (1 + √ 5)/2 ([18]).

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