Electronic Journal of Probability

On the Total External Length of the Kingman Coalescent

Svante Janson and Götz Kersting

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We prove asymptotic normality of the total length of external branches in the Kingman coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with black balls. Empty it step by step according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers of red balls form a Markov chain with an unexpected property: It is time-reversible.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 80, 2203-2218.

Accepted: 13 November 2011
First available in Project Euclid: 1 June 2016

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Zentralblatt MATH identifier

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)

coalescent external branch reversibility urn model

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Janson, Svante; Kersting, Götz. On the Total External Length of the Kingman Coalescent. Electron. J. Probab. 16 (2011), paper no. 80, 2203--2218. doi:10.1214/EJP.v16-955. https://projecteuclid.org/euclid.ejp/1464820249

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