Skip to main content

The Total External Branch Length of Beta-Coalescents


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]).

Hide All

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

Hide All
[1]Berestycki N. (2009) Recent progress in coalescent theory. Enasios Mathemáticos 16 1193.
[2]Berestycki J., Berestycki N. and Limic V. (2012) Asymptotic sampling formulae for Lambda-coalescents. To appear in Ann. Inst. H. Poincaré arXiv:1201.6512
[3]Berestycki J., Berestycki N. and Schweinsberg J. (2007) Beta-coalescents and continuous stable random trees. Ann. Probab. 35 18351887.
[4]Berestycki J., Berestycki N. and Schweinsberg J. (2008) Small time properties of Beta-coalescents. Ann. Inst. H. Poincaré 44 214238.
[5]Birkner M. and Blath J. (2008) Computing likelihoods for coalescents with multiple collisions in the infinitely-many-sites model. J. Math. Biology 57 435465.
[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 303325.
[7]Bolthausen E. and Sznitman A.-S. (1998) On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247276.
[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 16081614.
[9]Delmas J.-F., Dhersin J.-S. and Siri-Jégousse A. (2008) Asymptotic results on the length of coalescent trees. Ann. Appl. Probab. 18 9971025.
[10]Dhersin J.-S. and Yuan L. (2012) Asympotic behavior of the total length of external branches for Beta-coalescents. arXiv:1202.5859
[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 14041421.
[12]Durrett R. (2008) Probability Models for DNA Sequence Evolution, second edition, Springer.
[13]Eldon B. and Wakeley J. (2006) Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172 26212633.
[14]Gnedin A. and Yakubovich Y. (2007) On the number of collisions in Λ-coalescents. Electron. J. Probab. 12 15471567.
[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 2835.
[16]Iksanov A. and Möhle M. (2008) On the number of jumps of random walks with a barrier. Adv. Appl. Probab. 40 206228.
[17]Janson S. and Kersting G. (2011) On the total external length of the Kingman coalescent. Electron. J. Probab. 16 22032218.
[18]Kersting G. (2012) The asymptotic distribution of the length of Beta-coalescent trees. Ann. Appl. Probab. 22 20862107.
[19]Kingman J. F. C. (1982) The coalescent. Stoch. Proc. Appl. 13 235248.
[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 21592173.
[21]Pitman J. (1999) Coalescents with multiple collisions. Ann. Probab. 27 18701902.
[22]Sagitov S. (1999) The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 11161125.
[23]Schweinsberg J. (2000) A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Comm. Probab. 5 111.
[24]Steinrücken M., Birkner M. and Blath J. (2013) Analysis of DNA sequence variation within marine species using Beta-coalescents. Theoret. Popul. Biol. 83 2029.
[25]Wakeley J. (2008) Coalescent Theory: An Introduction, Roberts.
[26]Watterson G. A. (1975) On the number of segregating sites in genetical models without recombination. Theoret. Popul. Biol. 7 256276.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 40 *
Loading metrics...

Abstract views

Total abstract views: 187 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd November 2017. This data will be updated every 24 hours.