Annals of Applied Probability

Asymptotic results on the length of coalescent trees

Jean-François Delmas, Jean-Stéphane Dhersin, and Arno Siri-Jegousse

Full-text: Open access


We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natural estimator of DNA mutation rate for species with large families.

Article information

Ann. Appl. Probab., Volume 18, Number 3 (2008), 997-1025.

First available in Project Euclid: 26 May 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G52: Stable processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 05C05: Trees

Coalescent process Beta-coalescent stable process Watterson estimator


Delmas, Jean-François; Dhersin, Jean-Stéphane; Siri-Jegousse, Arno. Asymptotic results on the length of coalescent trees. Ann. Appl. Probab. 18 (2008), no. 3, 997--1025. doi:10.1214/07-AAP476.

Export citation


  • [1] Basdevant, A.-L. and Goldschmidt, C. (2007). Asymptotics of the allele frequency spectrum associated with the Bolthausen–Sznitman coalescent. Available at
  • [2] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007). Beta-coalescents and continuous stable random trees. Ann. Probab. 35 1835–1887.
  • [3] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007). Small time properties of Beta-coalescents. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [4] Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • [5] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147–181.
  • [6] Birkner, M., Blath, J., Capaldo, M., Etheridge, A., Möhle, M., Schweinsberg, J. and Wakolbinger, A. (2005). Alpha-stable branching and Beta-coalescents. Electron. J. Probab. 10 303–325.
  • [7] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276.
  • [8] Boom, J. D. G., Boulding, E. G. and Beckenbach, A. T. (1994). Mitochondrial DNA variation in introduced populations of pacific oyster, Crassostrea Gigas, in British Columbia. Can. J. Fish. Aquat. Sci. 51 1608–1614.
  • [9] Breiman, L. (1992). Probability. SIAM, Philadelphia.
  • [10] Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2007). Asymptotic results about the total branch length of the Bolthausen–Sznitman coalescent. Stoch. Process. Appl. 117 1404–1421.
  • [11] Eldon, B. and Wakeley, J. (2006). Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172 2621–2633.
  • [12] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. II. Wiley, New York.
  • [13] Gnedin, A. and Yakubovich, Y. (2007). On the number of collisions in Λ-coalescents. Available at
  • [14] Iksanov, A. and Möhle, M. (2007). On a random recursion related to absorption times of death Markov chains. Available at
  • [15] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [16] Kimura, M. (1969). The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations. Genetics 61 893–903.
  • [17] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248.
  • [18] Kingman, J. F. C. (2000). Origins of the coalescent 1974–1982. Genetics 156 1461–1463.
  • [19] Möhle, M. (2006). On the number of segregating sites for populations with large family sizes. Adv. in Appl. Probab. 38 750–767.
  • [20] Mukherjea, A., Rao, M. and Suen, S. (2006). A note on moment generating functions. Statist. Probab. Lett. 76 1185–1189.
  • [21] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
  • [22] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.
  • [23] Schweinsberg, J. (2003). Coalescent processes obtained from super critical Galton–Watson processes. Stochastic Process. Appl. 106 107–139.
  • [24] Watterson, G. A. (1975). On the number of segregating sites in genetical models without recombination. Theoret. Population Biology 7 256–276.