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Total internal and external lengths of the Bolthausen-Sznitman coalescent

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  30 March 2016

Götz Kersting ,
Juan Carlos Pardo  and
Arno Siri-Jégousse
Götz Kersting
Affiliation:
Goethe Universität, Robert Mayer Strasse 10, D-60325 Frankfurt am Main, Germany. Email address: [email protected].
Juan Carlos Pardo
Affiliation:
Centro de Investigación en Matemáticas (CIMAT), A.C., Calle Jalisco s/n, Col. Mineral de Valenciana, 36240 Guanajuato, Guanajuato, Mexico. Email address: [email protected].
Arno Siri-Jégousse
Affiliation:
Centro de Investigación en Matemáticas (CIMAT), A.C., Calle Jalisco s/n, Col. Mineral de Valenciana, 36240 Guanajuato, Guanajuato, Mexico. Email address: [email protected].
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Abstract

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In this paper we study a weak law of large numbers for the total internal length of the Bolthausen-Sznitman coalescent, thereby obtaining the weak limit law of the centered and rescaled total external length; this extends results obtained in Dhersin and Möhle (2013). An application to population genetics dealing with the total number of mutations in the genealogical tree is also given.

Keywords

Coalescent processBolthausen-Sznitman coalescentexternal branchblock counting processrecursive constructionIksanov-Möhle coupling

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J25: Continuous-time Markov processes on general state spaces 60F05: Central limit and other weak theorems 92D25: Population dynamics (general)
Type
Part 3. Biological applications
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 73 - 86
Copyright
Copyright © Applied Probability Trust 2014 

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