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The Total External Branch Length of Beta-Coalescents

  • IULIA DAHMER (a1), GÖTZ KERSTING (a1) and ANTON WAKOLBINGER (a1)
Abstract

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]).

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

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3] J. Berestycki , N. Berestycki and J. Schweinsberg (2007) Beta-coalescents and continuous stable random trees. Ann. Probab. 35 18351887.

[4] J. Berestycki , N. Berestycki and J. Schweinsberg (2008) Small time properties of Beta-coalescents. Ann. Inst. H. Poincaré 44 214238.

[5] M. Birkner and J. Blath (2008) Computing likelihoods for coalescents with multiple collisions in the infinitely-many-sites model. J. Math. Biology 57 435465.

[6] M. Birkner , J. Blath , M. Capaldo , A. Etheridge , M. Möhle , J. Schweinsberg and A. Wakolbinger (2005) Alpha-stable branching and Beta-coalescents. Electron. J. Probab. 10 303325.

[7] E. Bolthausen and A.-S. Sznitman (1998) On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247276.

[8] E. G. Boom , J. D. G. Boulding and A. T. Beckenbach (1994) Mitochondrial DNA variation in introduced populations of Pacific oyster, Crassostrea gigas, in British Columbia. Canad. J. Fish. Aquat. Sci. 51 16081614.

[9] J.-F. Delmas , J.-S. Dhersin and A. Siri-Jégousse (2008) Asymptotic results on the length of coalescent trees. Ann. Appl. Probab. 18 9971025.

[11] M. Drmota , A. Iksanov , M. Möhle and U. Rösler (2007) Asymptotic results about the total branch length of the Bolthausen–Sznitman coalescent. Stoch. Proc. Appl. 117 14041421.

[12] R. Durrett (2008) Probability Models for DNA Sequence Evolution, second edition, Springer.

[13] B. Eldon and J. Wakeley (2006) Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172 26212633.

[14] A. Gnedin and Y. Yakubovich (2007) On the number of collisions in Λ-coalescents. Electron. J. Probab. 12 15471567.

[15] A. Iksanov and M. Möhle (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] A. Iksanov and M. Möhle (2008) On the number of jumps of random walks with a barrier. Adv. Appl. Probab. 40 206228.

[17] S. Janson and G. Kersting (2011) On the total external length of the Kingman coalescent. Electron. J. Probab. 16 22032218.

[18] G. Kersting (2012) The asymptotic distribution of the length of Beta-coalescent trees. Ann. Appl. Probab. 22 20862107.

[19] J. F. C. Kingman (1982) The coalescent. Stoch. Proc. Appl. 13 235248.

[20] M. Möhle (2010) Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent. Stoch. Process. Appl. 120 21592173.

[21] J. Pitman (1999) Coalescents with multiple collisions. Ann. Probab. 27 18701902.

[26] G. A. Watterson (1975) On the number of segregating sites in genetical models without recombination. Theoret. Popul. Biol. 7 256276.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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