On the length distribution of external branches in coalescence trees: Genetic diversity within species

Let ZnZn denote the length of an external branch, chosen at random from a Kingman n  -coalescent. Based on a recursion for the distribution of ZnZn, we show that nZnnZn converges in distribution, as n tends to infinity, to a non-negative random variable Z   with density x↦8/(2+x)3x↦8/(2+x)3, x⩾0x⩾0.This result facilitates the study of the time to the most recent common ancestor of a randomly chosen individual and its closest relative in a given population. This time span also reflects the maximum relatedness between a single individual and the rest of the population. Therefore, it measures the uniqueness of a random individual, a central characteristic of the genetic diversity of a population.