Electronic Journal of Probability

On the Two Oldest Families for the Wright-Fisher Process

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

Full-text: Open access


We extend some of the results of Pfaffelhuber and Wakolbinger on the process of the most recent common ancestors in evolving coalescent by taking into account the size of one of the two oldest families or the oldest family which contains the immortal line of descent. For example we give an explicit formula for the Laplace transform of the extinction time for the Wright-Fisher diffusion. We give also an interpretation of the quasi-stationary distribution of the Wright-Fisher diffusion using the process of the relative size of one of the two oldest families, which can be seen as a resurrected Wright-Fisher diffusion.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 26, 776-800.

Accepted: 4 June 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92D25: Population dynamics (general)

Wright-Fisher diffusion MRCA Kingman coalescent tree resurrected process quasi-stationary distribution

This work is licensed under aCreative Commons Attribution 3.0 License.


Delmas, Jean-François; Dhersin, Jean-Stéphane; Siri-Jegousse, Arno. On the Two Oldest Families for the Wright-Fisher Process. Electron. J. Probab. 15 (2010), paper no. 26, 776--800. doi:10.1214/EJP.v15-771. https://projecteuclid.org/euclid.ejp/1464819811

Export citation


  • Brémaud, Pierre; Kannurpatti, Raghavan; Mazumdar, Ravi. Event and time averages: a review. Adv. in Appl. Probab. 24 (1992), no. 2, 377–411.
  • Cannings, C. The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Advances in Appl. Probability 6 (1974), 260–290.
  • Cattiaux, Patrick; Collet, Pierre; Lambert, Amaury; Martínez, Servet; Méléard, Sylvie; San Martín, Jaime. Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37 (2009), no. 5, 1926–1969.
  • Chang, Joseph T. Recent common ancestors of all present-day individuals. With discussion and reply by the author. Adv. in Appl. Probab. 31 (1999), no. 4, 1002–1038.
  • Collet, Pierre; Martínez, Servet; Maume-Deschamps, Véronique. On the existence of conditionally invariant probability measures in dynamical systems. Nonlinearity 13 (2000), no. 4, 1263–1274.
  • Darroch, J. N.; Seneta, E. On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probability 2 1965 88–100.
  • Donnelly, P.; Kurtz, T. G. The Eve process. Manuscript, personal communication.
  • Donnelly, Peter; Kurtz, Thomas G. A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24 (1996), no. 2, 698–742.
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166–205.
  • Durrett, Richard. Probability models for DNA sequence evolution. Second edition. Probability and its Applications (New York). Springer, New York, 2008. xii+431 pp. ISBN: 978-0-387-78168-6
  • Etheridge, Alison M. An introduction to superprocesses. University Lecture Series, 20. American Mathematical Society, Providence, RI, 2000. xii+187 pp. ISBN: 0-8218-2706-5
  • Evans, Steven N.; Ralph, Peter L. Dynamics of the time to the most recent common ancestor in a large branching population. Ann. Appl. Probab. 20 (2010), no. 1, 1–25.
  • Ewens, Warren J. Mathematical population genetics. I. Theoretical introduction. Second edition. Interdisciplinary Applied Mathematics, 27. Springer-Verlag, New York, 2004. xx+417 pp. ISBN: 0-387-20191-2
  • Ferrari, P. A.; Kesten, H.; Martinez, S.; Picco, P. Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab. 23 (1995), no. 2, 501–521.
  • Fisher, R. A. The genetical theory of natural selection. Revised reprint of the 1930 original. Oxford University Press, Oxford, 1999. xxii+332 pp. ISBN: 0-19-850440-3
  • Fu, Y. Exact coalescent for the Wright–Fisher model. Theoret. Population Biol. 69 (2006), no. 4, 385–394.
  • Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Convergence in distribution of random metric measure spaces ($Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 (2009), no. 1-2, 285–322.
  • Griffiths, R. C. Lines of descent in the diffusion approximation of neutral Wright-Fisher models. Theoret. Population Biol. 17 (1980), no. 1, 37–50.
  • Huillet, T. On Wright-Fisher diffusion and its relatives. J. Stat. Mech.: Theory and Experiment 11 (2007), ID P11006.
  • Johnson, Norman L.; Kotz, Samuel. Urn models and their application. An approach to modern discrete probability theory. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-London-Sydney, 1977. xiii+402 pp.
  • Kimura, M.; Ohta, T. The average number of generations until extinction of an individual mutant gene in a finite population. Genetics 63 (1969), no. 3, 701–709.
  • Kimura, M.; Ohta, T. The average number of generations until fixation of a mutant gene in a finite population. Genetics 61 (1969), no. 3, 763–771.
  • Kingman, J. F. C. Exchangeability and the evolution of large populations. Exchangeability in probability and statistics (Rome, 1981), pp. 97–112, North-Holland, Amsterdam-New York, 1982.
  • Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6
  • Möhle, Martin; Sagitov, Serik. A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 (2001), no. 4, 1547–1562.
  • Moran, P. A. P. Random processes in genetics. Proc. Cambridge Philos. Soc. 54 1958 60–71. (23 #B1034)
  • Pfaffelhuber, P.; Wakolbinger, A. The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 (2006), no. 12, 1836–1859.
  • Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870–1902.
  • Sagitov, Serik. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999), no. 4, 1116–1125.
  • Schweinsberg, Jason. Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 (2000), Paper no. 12, 50 pp. (electronic).
  • Simon, D.; Derrida, B. Evolution of the most recent common ancestor of a population with no selection. J. Stat. Mech.: Theory and Experiment 5 (2006), ID P05002.
  • Steinsaltz, David; Evans, Steven N. Quasistationary distributions for one-dimensional diffusions with killing. Trans. Amer. Math. Soc. 359 (2007), no. 3, 1285–1324 (electronic).
  • Tajima, F. Relationship between DNA polymorphism and fixation time. Genetics 125 (1990), no. 2, 447–454.
  • Wiuf, C.;Donnelly, P. Conditional genealogies and the age of a neutral mutant. Theoret. Population Biol. 56 (1999), 183–2001.
  • Wright, S. Evolution in Mendelian populations. Genetics 16 (1931), 97–159.