Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T01:15:17.792Z Has data issue: false hasContentIssue false
  • English
  • Français

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: kersting@math.uni-frankfurt.de.
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: jcpardo@cimat.mx.
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: arno@cimat.mx.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Basdevant, A.-L., and Goldschmidt, C. (2008). Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent. Electron. J. Prob. 13, 486–512.CrossRefGoogle Scholar
Berestycki, N. (2009). Recent Progress In Coalescent Theory (Math. Surveys 16). Sociedade Brasileira de Matemática, Rio de Janeiro.Google Scholar
Berestycki, J., Berestycki, N., and Schweinsberg, J. (2007). Beta-coalescents and continuous stable random trees. Ann. Prob. 35, 1835–1887.Google Scholar
Berestycki, J., Berestycki, N., and Schweinsberg, J. (2008). Small-time behavior of Beta-coalescents. Ann. Inst. H. Poincaré Prob. Statist. 44, 214–238.CrossRefGoogle Scholar
Berestycki, J., Berestycki, N., and Schweinsberg, J. (2013). The genealogy of branching Brownian motion with absorption. Ann. Prob. 41, 527–618.CrossRefGoogle Scholar
Bolthausen, E., and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.Google Scholar
Brunet, É., Derrida, B., Müller, A. H., and Munier, S. (2006). Noisy travelling waves: effect of selection on genealogies. Europhys. Lett. 76, 1–7.Google Scholar
Brunet, É., Derrida, B., Müller, A. H., and Munier, S. (2007). Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E 76, 041104, 20 pp.Google Scholar
Dahmer, I., Kersting, G., and Wakolbinger, A. (2014). The total external branch length of beta-coalescents. Combinatorics Prob. Comput. 23, 1010–1027.Google Scholar
Dhersin, J.-S. and Möhle, M. (2013). On the external branches of coalescents with multiple collisions. Electron. J. Prob. 18, 1–11.CrossRefGoogle Scholar
Dhersin, J.-S., and Yuan, L. (2015). On the total length of external branches for beta-coalescents. To appear in Adv. Appl. Prob.Google Scholar
Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2007). Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent. Stoch. Process. Appl. 117, 1404–1421.Google Scholar
Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2009). A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34, 319–336.Google Scholar
Geluk, J. L., and de Haan, L. (2000). Stable probability distributions and their domains of attraction: a direct approach. Prob. Math. Statist. 20, 169–188.Google Scholar
Iksanov, A. and Möhle, M. (2007). A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Commun. Prob. 12, 28–35.Google Scholar
Janson, S., and Kersting, G. (2011). On the total external length of the Kingman coalescent. Electron. J. Prob. 16, 2203–2218.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Möhle, M. (2010). Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent. Stoch. Process. Appl. 120, 2159–2173.Google Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.Google Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.Google Scholar