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

Let Zn denote the length of an external branch, chosen at random from a Kingman n-coalescent. Based on a recursion for the distribution of Zn, we show that nZn converges in distribution, as n tends to infinity, to a non-negative random variable Z with density x 8/(2+x)3, x 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.