Electronic Journal of Probability

Dynamics of the evolving Bolthausen-Sznitman coalecent

Jason Schweinsberg

Full-text: Open access


Consider a population of fixed size that evolves over time. At each time, the genealogical structure of the population can be described by a coalescent tree whose branches are traced back to the most recent common ancestor of the population. As time goes forward, the genealogy of the population evolves, leading to what is known as an evolving coalescent.  We will study the evolving coalescent for populations whose genealogy can be described by the Bolthausen Sznitman coalescent. We obtain the limiting behavior of the evolution of the time back to the most recent common ancestor and the total length of the branches in the tree. By similar methods, we also obtain a new result concerning the number of blocks in the Bolthausen-Sznitman coalescent.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 91, 50 pp.

Accepted: 16 October 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes 60G52: Stable processes 60G55: Point processes 92D25: Population dynamics (general)

Bolthausen-Sznitman coalescent most recent common ancestor total branch length

This work is licensed under aCreative Commons Attribution 3.0 License.


Schweinsberg, Jason. Dynamics of the evolving Bolthausen-Sznitman coalecent. Electron. J. Probab. 17 (2012), paper no. 91, 50 pp. doi:10.1214/EJP.v17-2378. https://projecteuclid.org/euclid.ejp/1465062413

Export citation


  • Berestycki, Julien; Berestycki, Nathanaël; Limic, Vlada. The $\Lambda$-coalescent speed of coming down from infinity. Ann. Probab. 38 (2010), no. 1, 207–233.
  • Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Small-time behavior of beta coalescents. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 2, 214–238.
  • Berestycki, J., Berestycki, N., and Schweinsberg, J.: The genealogy of branching Brownian motion with absorption, arXiv:1001.2337
  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0.
  • Bertoin, Jean; Le Gall, Jean-François. The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2000), no. 2, 249–266.
  • Bolthausen, E.; Sznitman, A.-S. On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 (1998), no. 2, 247–276.
  • Bovier, Anton; Kurkova, Irina. Much ado about Derrida's GREM. Spin glasses, 81–115, Lecture Notes in Math., 1900, Springer, Berlin, 2007.
  • Brunet, É.; Derrida, B.; Mueller, A. H.; Munier, S. Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76 (2007), no. 4, 041104, 20 pp.
  • Davis, M. H. A. Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. With discussion. J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353–388.
  • Davis, M. H. A. Markov models and optimization. Monographs on Statistics and Applied Probability, 49. Chapman & Hall, London, 1993. xiv+295 pp. ISBN: 0-412-31410-X.
  • Depperschmidt, A., Greven, A., and Pfaffelhuber, P.: Tree-valued Fleming-Viot dyamics with mutation and selection, arXiv:1101.0759
  • Delmas, Jean-François; Dhersin, Jean-Stéphane; Siri-Jegousse, Arno. On the two oldest families for the Wright-Fisher process. Electron. J. Probab. 15 (2010), no. 26, 776–800.
  • Drmota, Michael; Iksanov, Alex; Moehle, Martin; Roesler, Uwe. Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent. Stochastic Process. Appl. 117 (2007), no. 10, 1404–1421.
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166–205.
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
  • Evans, Steven N.; Pitman, Jim; Winter, Anita. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 (2006), no. 1, 81–126.
  • Evans, Steven N.; Ralph, Peter L. Dynamics of the time to the most recent common ancestor in a large branching population. Ann. Appl. Probab. 20 (2010), no. 1, 1–25.
  • Goldschmidt, Christina; Martin, James B. Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10 (2005), no. 21, 718–745 (electronic).
  • Greven, A., Pfaffelhuber, P., and Winter, A.: Tree-valued resampling dynamics: Martingale problems and applications, 0806.2224
  • Kingman, J. F. C. The coalescent. Stochastic Process. Appl. 13 (1982), no. 3, 235–248.
  • Klebaner, Fima C. Introduction to stochastic calculus with applications. Second edition. Imperial College Press, London, 2005. xiv+416 pp. ISBN: 1-86094-566-X.
  • Moran, P. A. P. Random processes in genetics. Proc. Cambridge Philos. Soc. 54 1958 60–71.
  • Pfaffelhuber, P.; Wakolbinger, A. The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 (2006), no. 12, 1836–1859.
  • Pfaffelhuber, P.; Wakolbinger, A.; Weisshaupt, H. The tree length of an evolving coalescent. Probab. Theory Related Fields 151 (2011), no. 3-4, 529–557.
  • Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870–1902.
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • Protter, Philip. Stochastic integration and differential equations. A new approach. Applications of Mathematics (New York), 21. Springer-Verlag, Berlin, 1990. x+302 pp. ISBN: 3-540-50996-8
  • Sagitov, Serik. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999), no. 4, 1116–1125.
  • Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4
  • Schweinsberg, Jason. Coalescent processes obtained from supercritical Galton-Watson processes. Stochastic Process. Appl. 106 (2003), no. 1, 107–139.
  • Simon, D. and Derrida, B.: Evolution of the most recent common ancestor of a population with no selection. J. Stat. Mech. Theory Exp., (2006), P05002.