The Annals of Applied Probability

The internal branch lengths of the Kingman coalescent

Iulia Dahmer and Götz Kersting

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In the Kingman coalescent tree the length of order $r$ is defined as the sum of the lengths of all branches that support $r$ leaves. For $r=1$ these branches are external, while for $r\ge2$ they are internal and carry a subtree with $r$ leaves. In this paper we prove that for any $s\in\mathbb{N}$ the vector of rescaled lengths of orders $1\le r\le s$ converges to the multivariate standard normal distribution as the number of leaves of the Kingman coalescent tends to infinity. To this end we use a coupling argument which shows that for any $r\ge2$ the (internal) length of order $r$ behaves asymptotically in the same way as the length of order 1 (i.e., the external length).

Article information

Ann. Appl. Probab. Volume 25, Number 3 (2015), 1325-1348.

First available in Project Euclid: 23 March 2015

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Coalescent internal branch length asymptotic distribution coupling Markov chain


Dahmer, Iulia; Kersting, Götz. The internal branch lengths of the Kingman coalescent. Ann. Appl. Probab. 25 (2015), no. 3, 1325--1348. doi:10.1214/14-AAP1024.

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  • [1] Basdevant, A.-L. and Goldschmidt, C. (2008). Asymptotics of the allele frequency spectrum associated with the Bolthausen–Sznitman coalescent. Electron. J. Probab. 13 486–512.
  • [2] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007). Beta-coalescents and continuous stable random trees. Ann. Probab. 35 1835–1887.
  • [3] Berestycki, N. (2009). Recent Progress in Coalescent Theory. Ensaios Matemáticos [Mathematical Surveys] 16. Sociedade Brasileira de Matemática, Rio de Janeiro.
  • [4] Caliebe, A., Neininger, R., Krawczak, M. and Rösler, U. (2007). On the length distribution of external branches in coalescent trees: Genetic diversity within species. Theor. Popul. Biol. 72 245–252.
  • [5] Dahmer, I., Kersting, G. and Wakolbinger, A. (2014). The total external branch length of Beta-coalescents. Combin. Probab. Comput. 23 1010–1027.
  • [6] Dhersin, J.-S. and Yuan, L. (2012). Asympotic behavior of the total length of external branches for Beta-coalescents. Available at arXiv:1202.5859.
  • [7] Durrett, R. (2008). Probability Models for DNA Sequence Evolution, 2nd ed. Springer, New York.
  • [8] Freund, F. and Möhle, M. (2009). On the time back to the most recent common ancestor and the external branch length of the Bolthausen–Sznitman coalescent. Markov Process. Related Fields 15 387–416.
  • [9] Fu, X. Y. (1995). Statistical properties of segregating sites. Theor. Popul. Biol. 48 172–197.
  • [10] Fu, Y. X. and Li, W. H. (1993). Statistical tests of neutrality of mutations. Genetics 133 693–709.
  • [11] Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Probab. 45 1186–1195.
  • [12] Janson, S. and Kersting, G. (2011). On the total external length of the Kingman coalescent. Electron. J. Probab. 16 2203–2218.
  • [13] Kersting, G., Pardo, J. C. and Siri-Jégousse, A. (2014). Total internal and external lengths of the Bolthausen–Sznitman coalescent. J. Appl. Probab. 51A (Celebrating 50 Years of Applied Probability Trust). To appear.
  • [14] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248.
  • [15] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [16] Möhle, M. (2010). Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent. Stochastic Process. Appl. 120 2159–2173.
  • [17] Wakeley, J. (2008). Coalescent Theory: An Introduction. Roberts & Company, Greenwood Village, CO.