The Annals of Applied Probability

The internal branch lengths of the Kingman coalescent

Iulia Dahmer and Götz Kersting

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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.

Export citation


  • [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.