Open Access
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 r2 they are internal and carry a subtree with r leaves. In this paper we prove that for any sN the vector of rescaled lengths of orders 1rs 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 r2 the (internal) length of order r behaves asymptotically in the same way as the length of order 1 (i.e., the external length).

Citation

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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

Keywords: asymptotic distribution , Coalescent , coupling , internal branch length , Markov chain

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.25 • No. 3 • June 2015
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