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The total external branch length of Beta-coalescents

For $1<α<2$ we derive the asymptotic distribution of the total length of {\em external} branches of a Beta$(2-α, α)$-coalescent as the number $n$ of leaves becomes large. It turns out the fluctuations of the external branch length follow those of $τ_n^{2-α}$ over the entire parameter regime, where $τ_n$ 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 $α_0 = (1+\sqrt 5)/2$.