Electronic Journal of Probability

On the external branches of coalescents with multiple collisions

Jean-Stéphane Dhersin and Martin Möhle

Full-text: Open access

Abstract

A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (Lambda-coalescents) is provided. This recursion is used to derive asymptotic results as the sample size n tends to infinity for the joint moments of the external branch lengths and for the moments of the total external branch length of the Bolthausen-Sznitman coalescent. These asymptotic results are based on a differential equation approach, which is as well useful to obtain exact solutions for the joint moments of the external branch lengths for the Bolthausen-Sznitman coalescent. The results for example show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen-Sznitman coalescent, whereas they are negatively correlated for the Kingman coalescent provided that n >= 4.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 40, 11 pp.

Dates
Accepted: 20 March 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064265

Digital Object Identifier
doi:10.1214/EJP.v18-2286

Mathematical Reviews number (MathSciNet)
MR3040550

Zentralblatt MATH identifier
1285.60079

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 34E05: Asymptotic expansions 60C05: Combinatorial probability 60J85: Applications of branching processes [See also 92Dxx] 92D15: Problems related to evolution 92D25: Population dynamics (general)

Keywords
Asymptotic expansions Bolthausen-Sznitman coalescent external branches joint moments Kingman coalescent multiple collisions

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Dhersin, Jean-Stéphane; Möhle, Martin. On the external branches of coalescents with multiple collisions. Electron. J. Probab. 18 (2013), paper no. 40, 11 pp. doi:10.1214/EJP.v18-2286. https://projecteuclid.org/euclid.ejp/1465064265


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