The internal branch lengths of the Kingman coalescent

TL;DR

This paper proves that the rescaled internal branch lengths of the Kingman coalescent converge to a multivariate normal distribution as the number of leaves grows large, using a coupling argument to relate internal and external lengths.

Contribution

It establishes the asymptotic normality of internal branch lengths in the Kingman coalescent and shows their behavior parallels external lengths for large trees.

Findings

Rescaled internal branch lengths converge to multivariate normal distribution.

Internal lengths behave asymptotically like external lengths.

Coupling argument links internal and external length distributions.

Abstract

In the Kingman coalescent tree the length of order $r$ is defined as the sum of the lengths of all branches that support $r$ leaves. For $r=1$ these branches are external, while for $r\ge2$ they are internal and carry a subtree with $r$ leaves. In this paper we prove that for any $s\in\mathbb{N}$ the vector of rescaled lengths of orders $1\le r\le s$ converges to the multivariate standard normal distribution as the number of leaves of the Kingman coalescent tends to infinity. To this end we use a coupling argument which shows that for any $r\ge2$ the (internal) length of order $r$ behaves asymptotically in the same way as the length of order 1 (i.e., the external length).