Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T12:56:00.775Z Has data issue: false hasContentIssue false
  • English
  • Français

The Total External Branch Length of Beta-Coalescents†

Published online by Cambridge University Press:  10 July 2014

IULIA DAHMER,
GÖTZ KERSTING and
ANTON WAKOLBINGER
IULIA DAHMER
Affiliation:
Institut für Mathematik, Goethe-Universität, 60054 Frankfurt am Main, Germany (e-mail: dahmer@math.uni-frankfurt.de, kersting@math.uni-frankfurt.de, wakolbinger@math.uni-frankfurt.de)
GÖTZ KERSTING
Affiliation:
Institut für Mathematik, Goethe-Universität, 60054 Frankfurt am Main, Germany (e-mail: dahmer@math.uni-frankfurt.de, kersting@math.uni-frankfurt.de, wakolbinger@math.uni-frankfurt.de)
ANTON WAKOLBINGER
Affiliation:
Institut für Mathematik, Goethe-Universität, 60054 Frankfurt am Main, Germany (e-mail: dahmer@math.uni-frankfurt.de, kersting@math.uni-frankfurt.de, wakolbinger@math.uni-frankfurt.de)
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.

For 1 < α < 2 we derive the asymptotic distribution of the total length of external branches of a Beta(2 − α, α)-coalescent as the number n of leaves becomes large. It turns out that the fluctuations of the external branch length follow those of τn2−α over the entire parameter regime, where τn denotes the random number of coalescences that bring the n lineages down to one. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at $\alpha_0 = (1+\sqrt 5)/2$ ([18]).

Keywords

Primary 60K35Secondary 60F0560J10

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014, pp. 1010 - 1027
Copyright
Copyright © Cambridge University Press 2014 

Footnotes

†

Work partially supported by the DFG Priority Programme SPP 1590 ‘Probabilistic Structures in Evolution’.

References

[1]Berestycki, N. (2009) Recent progress in coalescent theory. Enasios Mathemáticos 16 1–193.Google Scholar
[2]Berestycki, J., Berestycki, N. and Limic, V. (2012) Asymptotic sampling formulae for Lambda-coalescents. To appear in Ann. Inst. H. Poincaré arXiv:1201.6512Google Scholar
[3]Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007) Beta-coalescents and continuous stable random trees. Ann. Probab. 35 1835–1887.Google Scholar
[4]Berestycki, J., Berestycki, N. and Schweinsberg, J. (2008) Small time properties of Beta-coalescents. Ann. Inst. H. Poincaré 44 214–238.Google Scholar
[5]Birkner, M. and Blath, J. (2008) Computing likelihoods for coalescents with multiple collisions in the infinitely-many-sites model. J. Math. Biology 57 435–465.Google Scholar
[6]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.Google Scholar
[7]Bolthausen, E. and Sznitman, A.-S. (1998) On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276.Google Scholar
[8]Boom, E. G., Boulding, J. D. G. and Beckenbach, A. T. (1994) Mitochondrial DNA variation in introduced populations of Pacific oyster, Crassostrea gigas, in British Columbia. Canad. J. Fish. Aquat. Sci. 51 1608–1614.Google Scholar
[9]Delmas, J.-F., Dhersin, J.-S. and Siri-Jégousse, A. (2008) Asymptotic results on the length of coalescent trees. Ann. Appl. Probab. 18 997–1025.Google Scholar
[10]Dhersin, J.-S. and Yuan, L. (2012) Asympotic behavior of the total length of external branches for Beta-coalescents. arXiv:1202.5859Google Scholar
[11]Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2007) Asymptotic results about the total branch length of the Bolthausen–Sznitman coalescent. Stoch. Proc. Appl. 117 1404–1421.Google Scholar
[12]Durrett, R. (2008) Probability Models for DNA Sequence Evolution, second edition, Springer.Google Scholar
[13]Eldon, B. and Wakeley, J. (2006) Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172 2621–2633.Google Scholar
[14]Gnedin, A. and Yakubovich, Y. (2007) On the number of collisions in Λ-coalescents. Electron. J. Probab. 12 1547–1567.Google Scholar
[15]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. Comm. Probab. 12 28–35.CrossRefGoogle Scholar
[16]Iksanov, A. and Möhle, M. (2008) On the number of jumps of random walks with a barrier. Adv. Appl. Probab. 40 206–228.Google Scholar
[17]Janson, S. and Kersting, G. (2011) On the total external length of the Kingman coalescent. Electron. J. Probab. 16 2203–2218.Google Scholar
[18]Kersting, G. (2012) The asymptotic distribution of the length of Beta-coalescent trees. Ann. Appl. Probab. 22 2086–2107.Google Scholar
[19]Kingman, J. F. C. (1982) The coalescent. Stoch. Proc. Appl. 13 235–248.Google Scholar
[20]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
[21]Pitman, J. (1999) Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.Google Scholar
[22]Sagitov, S. (1999) The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.Google Scholar
[23]Schweinsberg, J. (2000) A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Comm. Probab. 5 1–11.Google Scholar
[24]Steinrücken, M., Birkner, M. and Blath, J. (2013) Analysis of DNA sequence variation within marine species using Beta-coalescents. Theoret. Popul. Biol. 83 20–29.Google Scholar
[25]Wakeley, J. (2008) Coalescent Theory: An Introduction, Roberts.Google Scholar
[26]Watterson, G. A. (1975) On the number of segregating sites in genetical models without recombination. Theoret. Popul. Biol. 7 256–276.Google Scholar