Journal of Applied Probability

Minimal clade size in the Bolthausen-Sznitman coalescent

Fabian Freund and Arno Siri-Jégousse

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In this article we show the asymptotics of distribution and moments of the size Xn of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman n-coalescent for n → ∞. The Bolthausen-Sznitman n-coalescent is a Markov process taking states in the set of partitions of {1, . . ., n}, where 1, . . ., n are referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in. We also provide exact formulae for the distribution of Xn. The main tool used is the connection of the Bolthausen-Sznitman n-coalescent with random recursive trees introduced by Goldschmidt and Martin (2005). With it, we show that Xn - 1 is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process with n - 1 customers.

Article information

J. Appl. Probab. Volume 51, Number 3 (2014), 657-668.

First available in Project Euclid: 5 September 2014

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs [See also 60B20] 60G09: Exchangeability 60F05: Central limit and other weak theorems 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)

Minimal clade size Bolthausen-Sznitman n-coalescent Chinese restaurant process


Freund, Fabian; Siri-Jégousse, Arno. Minimal clade size in the Bolthausen-Sznitman coalescent. J. Appl. Probab. 51 (2014), no. 3, 657--668. doi:10.1239/jap/1409932665.

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  • Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.
  • Berestycki, J., Berestycki, N. and Schweinsberg, J. (2013). The genealogy of branching Brownian motion with absorption. Ann. Prob. 41, 527–618.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.
  • Blum, M. G. B. and François, O. (2005). Minimal clade size and external branch length under the neutral coalescent. Adv. Appl. Prob. 37, 647–662.
  • Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.
  • Bovier, A. and Kurkova, I. (2007). Much ado about Derrida's GREM. In Spin Glasses (Lecture Notes Math. 1900), Springer, Berlin, pp. 81–115.
  • Brunet, É. and Derrida, B. (2013). Genealogies in simple models of evolution. J. Statist. Mech. Theory Exp. 2013, P01006.
  • Brunet, É., Derrida, B., Mueller, A. H. and Munier, S. (2006). Noisy traveling waves: effect of selection on genealogies. Europhys. Lett. 76, 1–7.
  • Brunet, É., Derrida, B., Mueller, A. H. and Munier, S. (2007). Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E 76, 041104.
  • Caliebe, A., Neininger, R., Krawczak, M. and Rösler, U. (2007). On the length distribution of external branches in coalescence trees: genetic diversity within species. Theoret. Pop. Biol. 72, 245–252.
  • DeLaurentis, J. M. and Pittel, B. G. (1985). Random permutations and Brownian motion. Pacific J. Math. 119, 287–301.
  • Desai, M. M., Walczak, A. M. and Fisher, D. S. (2013). Genetic diversity and the structure of genealogies in rapidly adapting populations. Genetics 193, 565–585.
  • Dhersin, J.-S., Freund, F., Siri-Jégousse, A. and Yuan, L. (2013). On the length of an external branch in the beta-coalescent. Stoch. Process. Appl 123, 1691–1715.
  • 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. Relat. Fields 15, 387–416.
  • Fu, Y. X. and Li, W. H. (1993). Statistical tests of neutrality of mutations. Genetics 133, 693–709.
  • Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Prob. 45, 1186–1195.
  • Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Prob. 10, 718–745.
  • Hansen, J. C. (1990). A functional central limit theorem for the Ewens sampling formula. J. Appl. Prob. 27, 28–43.
  • Hwang, H.-K. (1995). Asymptotic expansions for the Stirling numbers of the first kind. J. Combin. Theory A 71, 343–351.
  • Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.
  • Neher, R. and Hallatschek, O. (2013). Genealogies of rapidly adapting populations. Proc. Nat. Acad. Sci. USA 110, 437–442.
  • Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.
  • Pitman, J. (2005). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin.
  • Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.