On the Total External Length of the Kingman Coalescent

Svante Janson (Uppsala University)
Götz Kersting (Goethe university Frankfurt)


We prove asymptotic normality of the total length of external branches in the Kingman coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with black balls. Empty it step by step according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers of red balls form a Markov chain with an unexpected property: It is time-reversible.

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Pages: 2203-2218

Publication Date: November 13, 2011

DOI: 10.1214/EJP.v16-955


  1. J. Berestycki, N. Berestycki and J. Schweinsberg. Beta-coalescents and continuous stable random trees. Ann. Probab. 35 (2007), 1835-1887. Math. Review 2009d:60244
  2. J. Berestycki, N. Berestycki and V. Limic. Asymptotic sampling formulae and particle system representations for Lambda-coalescents. Preprint available at arXiv:1101.1875
  3. M.G.B. Blum and O. Francois. Minimal clade size and external branch length under the neutral coalescent. Adv. Appl. Prob. 37 (2005), 647-662. Math. Review 2006f:92020
  4. A. Caliebe, R. Neininger, M. Krawczak and U. Rösler. On the length distribution of external branches in coalescent trees: Genetic diversity within species. Theor. Population Biology. 72 (2007), 245-252.
  5. R. Durrett. Probability models for DNA sequence evolution. Springer-Verlag (2002) Berlin, New-York. Math. Review 2003b:60003
  6. F. Freund and M. Möhle. On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Proc. Rel. Fields. 15 (2009), 387-416. Math. Review 2011b:60296
  7. Y.X. Fu and W.H. Li. Statistical tests of neutrality of mutations. Genetics 133 (1993), 693-709.
  8. A. Gnedin, A. Iksanov and M. Möhle. On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Probab. 45 (2007), 1186-1195. Math. Review 2010b:60096
  9. S. Janson. Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence. Ann. Comb. 12 (2009), 417-447. Math. Review 2011a:60042
  10. S. Janson and J. Spencer. A point process describing the component sizes in the critical window of the random graph evolution. Combin. Probab. Comput. 16 (2007), 631-658. Math. Review 2008f:05179
  11. O. Kallenberg. Foundations of Modern Probability, second ed. Springer-Verlag (2002) Berlin, New-York. Math. Review 2003b:60003
  12. J.F.C. Kingman. The coalescent. Stoch. Process. Appl. 13 (1982), 235-248. Math. Review 84a:60079
  13. M. Möhle. Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent. Stoch. Process. Appl. 120 (2010), 2159-2173. Math. Review 2011j:60235
  14. J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999), 1870-1902. Math. Review 2001h:60016
  15. D. Steinsaltz. Random time changes for sock-sorting and other stochastic process limit theorems. Electron. J. Probab. 4 (1999), 1-25. Math. Review 2000e:60038

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