The Annals of Applied Probability

The fixation line in the ${\Lambda}$-coalescent

Olivier Hénard

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Abstract

We define a Markov process in a forward population model with backward genealogy given by the $\Lambda$-coalescent. This Markov process, called the fixation line, is related to the block counting process through its hitting times. Two applications are discussed. The probability that the $n$-coalescent is deeper than the $(n-1)$-coalescent is studied. The distribution of the number of blocks in the last coalescence of the $n$-$\operatorname{Beta}(2-\alpha,\alpha)$-coalescent is proved to converge as $n\rightarrow\infty$, and the generating function of the limiting random variable is computed.

Article information

Source
Ann. Appl. Probab. Volume 25, Number 5 (2015), 3007-3032.

Dates
Received: August 2013
Revised: September 2014
First available in Project Euclid: 30 July 2015

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1438261058

Digital Object Identifier
doi:10.1214/14-AAP1077

Mathematical Reviews number (MathSciNet)
MR3375893

Zentralblatt MATH identifier
1325.60124

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60G55: Point processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Coalescent Markov chain duality hitting times

Citation

Hénard, Olivier. The fixation line in the ${\Lambda}$-coalescent. Ann. Appl. Probab. 25 (2015), no. 5, 3007--3032. doi:10.1214/14-AAP1077. http://projecteuclid.org/euclid.aoap/1438261058.


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References

  • [1] Abraham, R. and Delmas, J.-F. (2013). $\beta$-coalescents and stable Galton–Watson trees. Preprint. Available at arXiv:1303.6882.
  • [2] Abraham, R. and Delmas, J.-F. (2013). A construction of a $\beta$-coalescent via the pruning of binary trees. J. Appl. Probab. 50 772–790.
  • [3] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2008). Small-time behavior of beta coalescents. Ann. Inst. Henri Poincaré Probab. Stat. 44 214–238.
  • [4] Berestycki, N. (2009). Recent Progress in Coalescent Theory. Ensaios Matemáticos [Mathematical Surveys] 16. Sociedade Brasileira de Matemática, Rio de Janeiro.
  • [5] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [6] Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 261–288.
  • [7] Birkner, M., Blath, J., Capaldo, M., Etheridge, A., Möhle, M., Schweinsberg, J. and Wakolbinger, A. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 303–325 (electronic).
  • [8] Caliebe, A., Neininger, R., Krawczak, M. and Rösler, U. (2007). On the length distribution of external branches in coalescence trees: Genetic diversity within species. Theor. Popul. Biol. 72 245–252.
  • [9] Dhersin, J.-S. and Möhle, M. (2013). On the external branches of coalescents with multiple collisions. Electron. J. Probab. 18 1–11.
  • [10] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
  • [11] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147.
  • [12] Gnedin, A., Iksanov, A. and Marynych, A. (2014). A survey of the $\Lambda$-coalescents. Preprint. Available at http://unicyb.kiev.ua/~iksan/publications/Lambda-Survey100814.pdf.
  • [13] Gnedin, A., Iksanov, A., Marynych, A. and Möhle, M. (2014). On asymptotics of the beta coalescents. Adv. in Appl. Probab. 46 496–515.
  • [14] Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Probab. 10 718–745 (electronic).
  • [15] Grey, D. R. (1977). Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Probab. 14 702–716.
  • [16] Harris, T. E. (1963). The Theory of Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 119. Springer, Berlin.
  • [17] Hénard, O. (2013). Change of measure in the lookdown particle system. Stochastic Process. Appl. 123 2054–2083.
  • [18] Kersting, G. (2012). The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Probab. 22 2086–2107.
  • [19] Labbé, C. (2014). From flows of $\Lambda$-Fleming–Viot processes to lookdown processes via flows of partitions. Electron. J. Probab. 19 1–49.
  • [20] Limic, V. (2011). Coalescent processes and reinforced random walks: A guide through martingales and coupling. Habilitation thesis. Available at http://www.math.u-psud.fr/~limic/izmars/habi.html.
  • [21] Möhle, M. (2014). Asymptotic hitting probabilities for the Bolthausen–Sznitman coalescent. J. Appl. Probab. A 51 87–97.
  • [22] Möhle, M. (2014). On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity. ALEA Lat. Am. J. Probab. Math. Stat. 11 141–159.
  • [23] Möhle, M. and Pitters, H. (2014). A spectral decomposition for the block counting process of the Bolthausen–Sznitman coalescent. Electron. Commun. Probab. 19 1–11.
  • [24] Neveu, J. (1992). A continuous-state branching process in relation with the GREM model of spin glass theory. Rapport interne 267, Ecole Polytechnique.
  • [25] Ostrowski, A. M. (1949). On some generalizations of the Cauchy–Frullani integral. Proc. Natl. Acad. Sci. USA 35 612–616.
  • [26] Pfaffelhuber, P. and Wakolbinger, A. (2006). The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 1836–1859.
  • [27] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
  • [28] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 1: Foundations. Cambridge Univ. Press, Cambridge. Reprint of the second (1994) edition.
  • [29] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.
  • [30] Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Commun. Probab. 5 1–11 (electronic).