Advertisement

A Note on the Small-Time Behaviour of the Largest Block Size of Beta n-Coalescents

  • Arno Siri-JégousseEmail author
  • Linglong Yuan
Conference paper
Part of the Progress in Probability book series (PRPR, volume 73)

Abstract

We study the largest block size of Beta n-coalescents at small times as n tends to infinity, using the paintbox construction of Beta-coalescents and the link between continuous-state branching processes and Beta-coalescents established in Birkner et al. (Electron J Probab 10(9):303–325, 2005) and Berestycki et al. (Ann Inst H Poincaré Probab Stat 44(2):214–238, 2008). As a corollary, a limit result on the largest block size at the coalescence time of the individual/block {1} is provided.

Keywords

Beta-coalescent Kingman’s paintbox construction Continuous-state branching processes Largest block size Block-counting process 

2010 Mathematics Subject Classification

60J25 60F05 92D15 

Notes

Acknowledgements

The authors would like to thank Fabian Freund for some very valuable comments. Arno Siri-Jégousse would like to welcome Raphaël Clodic Griffon, born the day the final version was sent. ASJ is supported by CONACyT Grant CB-2014/243068.

References

  1. 1.
    J. Berestycki, N. Berestycki, J. Schweinsberg, Beta-coalescents and continuous stable random trees. Ann. Probab. 35(5), 1835–1887 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Berestycki, N. Berestycki, J. Schweinsberg, Small-time behavior of Beta-coalescents. Ann. Inst. H. Poincaré Probab. Stat. 44(2), 214–238 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Bertoin, J.-F. Le Gall, The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Relat. Fields 117(2), 249–266 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Birkner, J. Blath, M. Capaldo, A.M. Etheridge, M. Möhle, J. Schweinsberg, A. Wakolbinger, Alpha-stable branching and Beta-coalescents. Electron. J. Probab. 10(9), 303–325 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M.G.B. Blum, O. François, Minimal clade size and external branch length under the neutral coalescent. Adv. Appl. Probab. 37(3), 647–662 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Caliebe, R. Neininger, M. Krawczak, U. Rösler, On the length distribution of external branches in coalescence trees: genetic diversity within species. Theor. Popul. Biol. 72(2), 245–252 (2007)CrossRefGoogle Scholar
  7. 7.
    I. Dahmer, G. Kersting, A. Wakolbinger, The total external branch length of Beta-coalescents. Comb. Probab. Comput. 23, 1–18 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.-F. Delmas, J.-S. Dhersin, A. Siri-Jégousse, Asymptotic results on the length of coalescent trees. Ann. Appl. Probab. 18(3), 997–1025 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    J.-S. Dhersin, F. Freund, A. Siri-Jégousse, L. Yuan, On the length of an external branch in the beta-coalescent. Stoch. Process. Appl. 123, 1691–1715 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Donnelly, T.G. Kurtz, Particle representations for measure-valued population models. Ann. Probab. 27(1), 166–205 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    F. Freund, A. Siri-Jégousse, Minimal clade size in the Bolthausen-Sznitman coalescent. J. Appl. Probab. 51(3), 657–668 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    G. Kersting, The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Probab. 22(5), 2086–2107 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    G. Kersting, J. Schweinsberg, A. Wakolbinger, The evolving beta coalescent. Electron. J. Probab. 19(64), 1–27 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    J.F.C. Kingman, The coalescent. Stoch. Process. Appl. 13(3), 235–248 (1982)MathSciNetCrossRefGoogle Scholar
  15. 15.
    L. Miller, H. Pitters, Small-time behaviour and hydrodynamic limits of beta coalescents. Preprint on Arxiv. https://arxiv.org/pdf/1611.06280.pdf
  16. 16.
    M. Möhle, S. Sagitov, A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29(4)(500), 1547–1562 (2001)Google Scholar
  17. 17.
    J. Pitman, Coalescents with multiple collisions. Ann. Probab. 27(4), 1870–1902 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    S. Sagitov, The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36(4), 1116–1125 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. Schweinsberg, A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Commun. Probab. 5, 1–11 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Schweinsberg, Coalescent processes obtained from supercritical Galton-Watson processes. Stoch. Process. Appl. 106(1), 107–139 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    B. Şengül, Asymptotic number of caterpillars of regularly varying Λ-coalescents that come down from infinity. Electron. Commun. Probab. 22 (2017)Google Scholar
  22. 22.
    A. Siri-Jégousse, L. Yuan, Asymptotics of the minimal clade size and related functionals of certain Beta-coalescents. Acta Appl. Math. 142(1), 127–148 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    R. Slack, A branching process with mean one and possibly infinite variance. Probab. Theory Relat. Fields 9(2), 139–145 (1968)MathSciNetzbMATHGoogle Scholar
  24. 24.
    L. Yuan, On the measure division construction of Λ-coalescents. Markov Process. Relat. Fields 20, 229–264 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IIMASUniversidad Nacional Autónoma de MéxicoMexico CityMexico
  2. 2.Department of Mathematical SciencesXi’an Jiaotong-Liverpool UniversitySuzhouPeople’s Republic of China

Personalised recommendations