## The Annals of Applied Probability

### The internal branch lengths of the Kingman coalescent

#### 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).

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1325-1348.

Dates
First available in Project Euclid: 23 March 2015

https://projecteuclid.org/euclid.aoap/1427124130

Digital Object Identifier
doi:10.1214/14-AAP1024

Mathematical Reviews number (MathSciNet)
MR3325275

Zentralblatt MATH identifier
1315.60107

#### Citation

Dahmer, Iulia; Kersting, Götz. The internal branch lengths of the Kingman coalescent. Ann. Appl. Probab. 25 (2015), no. 3, 1325--1348. doi:10.1214/14-AAP1024. https://projecteuclid.org/euclid.aoap/1427124130

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