The asymptotic distribution of the length of beta-coalescent trees

Abstract

We derive the asymptotic distribution of the total length Ln of a Beta(2- α,α)-coalescent tree for 1 < α < 2, starting from n individuals. There are two regimes: If α ≤ 12 (1 + √ 5), then Ln suitably rescaled has a stable limit distribution of index α. Otherwise Ln just has to be shifted by a constant (depending on n) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number Sn of segregation sites. These are points (mutations), which are placed on the tree's branches according to a Poisson point process with constant rate. © Institute of Mathematical Statistics, 2012.