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June 2015 The internal branch lengths of the Kingman coalescent
Iulia Dahmer, Götz Kersting
Ann. Appl. Probab. 25(3): 1325-1348 (June 2015). DOI: 10.1214/14-AAP1024

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).

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Iulia Dahmer. Götz Kersting. "The internal branch lengths of the Kingman coalescent." Ann. Appl. Probab. 25 (3) 1325 - 1348, June 2015. https://doi.org/10.1214/14-AAP1024

Information

Published: June 2015
First available in Project Euclid: 23 March 2015

zbMATH: 1315.60107
MathSciNet: MR3325275
Digital Object Identifier: 10.1214/14-AAP1024

Subjects:
Primary: 60K35
Secondary: 60F05, 60J10

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 3 • June 2015
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