The Annals of Applied Probability

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

Olivier Hénard

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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
https://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. https://projecteuclid.org/euclid.aoap/1438261058.


Export citation

References

  • [1] Abraham, R. and Delmas, J.-F. (2013). $\beta$-coalescents and stable Galton–Watson trees. Preprint. Available at arXiv:1303.6882.
    arXiv: 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.
    Mathematical Reviews (MathSciNet): MR3102514
    Zentralblatt MATH: 1285.60078
    Digital Object Identifier: doi:10.1239/jap/1378401235
    Project Euclid: euclid.jap/1378401235
  • [3] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2008). Small-time behavior of beta coalescents. Ann. Inst. Henri Poincaré Probab. Stat. 44 214–238.
    Mathematical Reviews (MathSciNet): MR2446321
    Zentralblatt MATH: 1214.60034
    Digital Object Identifier: doi:10.1214/07-AIHP103
    Project Euclid: euclid.aihp/1207948217
  • [4] Berestycki, N. (2009). Recent Progress in Coalescent Theory. Ensaios Matemáticos [Mathematical Surveys] 16. Sociedade Brasileira de Matemática, Rio de Janeiro.
    Mathematical Reviews (MathSciNet): MR2574323
    Zentralblatt MATH: 1204.60002
  • [5] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR2253162
  • [6] Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 261–288.
    Mathematical Reviews (MathSciNet): MR1990057
    Zentralblatt MATH: 1023.92018
    Digital Object Identifier: doi:10.1007/s00440-003-0264-4
  • [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).
    Mathematical Reviews (MathSciNet): MR2120246
    Zentralblatt MATH: 1066.60072
    Digital Object Identifier: doi:10.1214/EJP.v10-241
  • [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.
    Mathematical Reviews (MathSciNet): MR3040550
    Zentralblatt MATH: 1285.60079
    Digital Object Identifier: doi:10.1214/EJP.v18-2286
  • [10] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
    Mathematical Reviews (MathSciNet): MR1681126
    Zentralblatt MATH: 0956.60081
    Digital Object Identifier: doi:10.1214/aop/1022677258
    Project Euclid: euclid.aop/1022677258
  • [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.
    Mathematical Reviews (MathSciNet): MR3215543
    Zentralblatt MATH: 06316060
    Digital Object Identifier: doi:10.1239/aap/1401369704
    Project Euclid: euclid.aap/1401369704
  • [14] Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Probab. 10 718–745 (electronic).
    Mathematical Reviews (MathSciNet): MR2164028
    Zentralblatt MATH: 1109.60060
    Digital Object Identifier: doi:10.1214/EJP.v10-265
  • [15] Grey, D. R. (1977). Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Probab. 14 702–716.
    Mathematical Reviews (MathSciNet): MR478377
    Zentralblatt MATH: 0378.60064
    Digital Object Identifier: doi:10.2307/3213344
  • [16] Harris, T. E. (1963). The Theory of Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 119. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR163361
  • [17] Hénard, O. (2013). Change of measure in the lookdown particle system. Stochastic Process. Appl. 123 2054–2083.
    Mathematical Reviews (MathSciNet): MR3038498
    Zentralblatt MATH: 1302.60107
    Digital Object Identifier: doi:10.1016/j.spa.2013.01.015
  • [18] Kersting, G. (2012). The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Probab. 22 2086–2107.
    Mathematical Reviews (MathSciNet): MR3025690
    Zentralblatt MATH: 1251.92034
    Digital Object Identifier: doi:10.1214/11-AAP827
    Project Euclid: euclid.aoap/1350067995
  • [19] Labbé, C. (2014). From flows of $\Lambda$-Fleming–Viot processes to lookdown processes via flows of partitions. Electron. J. Probab. 19 1–49.
    Mathematical Reviews (MathSciNet): MR3227064
    Zentralblatt MATH: 06309119
  • [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.
    Mathematical Reviews (MathSciNet): MR3317352
    Zentralblatt MATH: 06424989
    Digital Object Identifier: doi:10.1239/jap/1417528469
    Project Euclid: euclid.jap/1417528469
  • [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.
    Mathematical Reviews (MathSciNet): MR3225970
    Zentralblatt MATH: 06290081
  • [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.
    Mathematical Reviews (MathSciNet): MR3246966
    Zentralblatt MATH: 06349205
  • [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.
    Mathematical Reviews (MathSciNet): MR31020
    Zentralblatt MATH: 0034.32201
    Digital Object Identifier: doi:10.1073/pnas.35.10.612
  • [26] Pfaffelhuber, P. and Wakolbinger, A. (2006). The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 1836–1859.
    Mathematical Reviews (MathSciNet): MR2307061
    Zentralblatt MATH: 1110.92030
    Digital Object Identifier: doi:10.1016/j.spa.2006.04.015
  • [27] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
    Mathematical Reviews (MathSciNet): MR1742892
    Zentralblatt MATH: 0963.60079
    Digital Object Identifier: doi:10.1214/aop/1022677552
    Project Euclid: euclid.aop/1022874819
  • [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.
    Mathematical Reviews (MathSciNet): MR1796539
  • [29] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.
    Mathematical Reviews (MathSciNet): MR1742154
    Zentralblatt MATH: 0962.92026
    Digital Object Identifier: doi:10.1239/jap/1032374759
    Project Euclid: euclid.jap/1032374759
  • [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).
    Mathematical Reviews (MathSciNet): MR1736720
    Zentralblatt MATH: 0953.60072

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more