External branch lengths of $\Lambda $-coalescents without a dust component

Abstract

$\Lambda $-coalescents model genealogies of samples of individuals from a large population by means of a family tree. The tree’s leaves represent the individuals, and the lengths of the adjacent edges indicate the individuals’ time durations up to some common ancestor. These edges are called external branches. We consider typical external branches under the broad assumption that the coalescent has no dust component and maximal external branches under further regularity assumptions. As it transpires, the crucial characteristic is the coalescent’s rate of decrease $\mu (b)$, $b\geq 2$. The magnitude of a typical external branch is asymptotically given by $n/\mu (n)$, where $n$ denotes the sample size. This result, in addition to the asymptotic independence of several typical external lengths, holds in full generality, while convergence in distribution of the scaled external lengths requires that $\mu (n)$ is regularly varying at infinity. For the maximal lengths, we distinguish two cases. Firstly, we analyze a class of $\Lambda $-coalescents coming down from infinity and with regularly varying $\mu $. Here, the scaled external lengths behave as the maximal values of $n$ i.i.d. random variables, and their limit is captured by a Poisson point process on the positive real line. Secondly, we turn to the Bolthausen-Sznitman coalescent, where the picture changes. Now, the limiting behavior of the normalized external lengths is given by a Cox point process, which can be expressed by a randomly shifted Poisson point process.