The Annals of Applied Probability

The internal branch lengths of the Kingman coalescent

Iulia Dahmer and Götz Kersting

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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

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1427124130

Digital Object Identifier
doi:10.1214/14-AAP1024

Mathematical Reviews number (MathSciNet)
MR3325275

Zentralblatt MATH identifier
1315.60107

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Coalescent internal branch length asymptotic distribution coupling Markov chain

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


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