Electronic Journal of Probability

On the Total External Length of the Kingman Coalescent

Svante Janson and Götz Kersting

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820249

Digital Object Identifier
doi:10.1214/EJP.v16-955

Mathematical Reviews number (MathSciNet)
MR2861672

Zentralblatt MATH identifier
1245.60094

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 external branch reversibility urn model

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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