Electronic Journal of Probability

Coalescent results for diploid exchangeable population models  

Matthias Birkner, Huili Liu, and Anja Sturm

Full-text: Open access


We consider diploid bi-parental analogues of Cannings models: in a population of fixed size $N$ the next generation is composed of $V_{i,j}$ offspring from parents $i$ and $j$, where $V=(V_{i,j})_{1\le i \neq j \le N}$ is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly scaled ancestral process for an $n$-sample of genes towards a ($\Xi $-)coalescent. This complements Möhle and Sagitov’s (2001) result for the haploid case and sharpens the profile of Möhle and Sagitov’s (2003) study of the diploid case, which focused on fixed couples, where each row of $V$ has at most one non-zero entry.

We apply the convergence result to several examples, in particular to two diploid variations of Schweinsberg’s (2003) model, leading to Beta-coalescents with two-fold and with four-fold mergers, respectively.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 49, 44 pp.

Received: 9 September 2017
Accepted: 30 April 2018
First available in Project Euclid: 1 June 2018

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60J17 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D25: Population dynamics (general)

diploid population model diploid ancestral process coalescent with simultaneous multiple collisions

Creative Commons Attribution 4.0 International License.


Birkner, Matthias; Liu, Huili; Sturm, Anja. Coalescent results for diploid exchangeable population models. Electron. J. Probab. 23 (2018), paper no. 49, 44 pp. doi:10.1214/18-EJP175. https://projecteuclid.org/euclid.ejp/1527818427

Export citation


  • [1] Austin, T.: On exchangeable random variables and the statistics of large graphs and hypergraphs. Probab. Surv. 5, (2008), 80–145.
  • [2] Berestycki, J.; Berestycki, N.; Schweinsberg, J.: Beta-coalescents and continuous stable random trees. Ann. Probab. 35, (2007), 1835–1887.
  • [3] Birkner, M.; Blath, J.; Capaldo, M.; Etheridge, A.; Möhle, M.; Schweinsberg J.; Wakolbinger. A.: Alpha-stable Branching and Beta-Coalescents. Electron. J. Probab. 10, (2005), 303–325.
  • [4] Birkner, M.; Blath, J.; Eldon, B.: An ancestral recombination graph for diploid populations with skewed offspring distribution. Genetics 193, (2013), 255–290. arXiv:1203.4950
  • [5] Billingsley, P.: Convergence of Probability Measures, 2nd edition. John Wiley & Sons, Inc., Chicago, 1999.
  • [6] Chvátal, V.: The tail of the hypergeometric distribution. Discrete Math. 25, (1979), 285–287.
  • [7] Dembo, A. and Zeitouni, O.: Large deviations techniques and applications. 2nd corrected edition. Springer-Verlag, Berlin, 2010. xvi+396 pp.
  • [8] Donnelly, P. and Kurtz, T. G.: Particle representations for measure-valued population models. Ann. Probab. 27, (1999), 166–205.
  • [9] Eldon, B. and Wakeley, J.: Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172, (2006), 2621–2633.
  • [10] Ethier, S. N. and Kurtz, T. G.: Markov processes: Characterization and convergence. John Wiley & Sons, Inc., New Jersey, 1986.
  • [11] Feller, W.: An Introduction to Probability Theory and Its Applications, II, 2nd edition. John Wiley & Sons, Inc., New York-London-Sydney, 1971. xxiv+669 pp.
  • [12] Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, (1963), 13–30.
  • [13] Johnson, N. L.; Kotz, S.; Balakrishnan, N.: Discrete Multivariate Distributions. John Wiley & Sons, Inc., New York, 1997. xxii+299 pp.
  • [14] Kallenberg, O.: Probabilistic symmetries and invariance principles. Springer, New York, 2005. xii+510 pp.
  • [15] Kingman, J. F. C.: On the genealogy of large populations. J. Appl. Probab. 19, (1982), 27–43.
  • [16] Kingman, J. F. C.: The coalescent. Stochastic Process. Appl. 13, (1982), 235–248.
  • [17] Möhle, M.: A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing. Adv. in Appl. Probab. 30, (1998), 493–512.
  • [18] Möhle, M.: Coalescent results for two-sex population models. Adv. in Appl. Probab. 30, (1998), 513–520.
  • [19] Möhle, M.: Weak convergence to the coalescent in neutral population models. J. Appl. Probab. 36, (1999), 446–460.
  • [20] Möhle, M.: Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. in Appl. Probab. 32, (2000), 983–993.
  • [21] Möhle, M.: On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12, (2006), 35-53.
  • [22] Möhle, M. and Sagitov, S.: A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29, (2001), 1547–1562.
  • [23] Möhle, M. and Sagitov, S.: Coalescent patterns in diploid exchangeable population models. J. Math. Biol. 47, (2003), 337–352.
  • [24] Nagaev, S. V.: On the asymptotic behavior of probabilities of one-sided large deviations. Teor. Veroyatnost. i Primenen 26, (1981), 369–372.
  • [25] Pitman, J.: Coalescents with multiple collisions. Ann. Probab. 27, (1999), 1870-1902.
  • [26] Sagitov, S.: The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36, (1999), 1116-1125.
  • [27] Sagitov, S.: Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Probab. 40, (2003), 839-854.
  • [28] Schweinsberg, J.: Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5, (2000), 1-50.
  • [29] Schweinsberg, J.: Coalescent processes obtained from supercritical Galton-Watson processes. Stochastic Process. Appl. 106, (2003), 107-139.
  • [30] van der Hofstad, R.: Random Graphs and Complex Networks, Vol. I and II. Cambridge University Press, 2017. http://www.win.tue.nl/ rhofstad/NotesRGCN.html