Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T15:35:26.269Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence to the coalescent with simultaneous multiple mergers

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Serik Sagitov
Serik Sagitov*
Affiliation:
Chalmers University of Technology
*
∗Postal address: School of Mathematical Sciences, Chalmers University of Technology, S-412 96 Göteborg, Sweden. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized by Möhle and Sagitov (2001) in terms of a sequence of measures defined on the finite-dimensional simplices. A more compact characterization of the general coalescent requiring a single probability measure Ξ defined on the infinite simplex Δ was suggested by Schweinsberg (2000). This paper presents a simple criterion of weak convergence to the Ξ-coalescent. In contrast to the earlier criterion of Möhle and Sagitov (2001) based on the moment conditions, the key condition here is expressed in terms of the joint distribution of the ranked offspring sizes. This criterion interprets a vector in Δ as the ranked fractions of the total population size assigned to sibling groups constituting a (rare) generation, where a merger might occur. An example of the general coalescent is developed on the basis of the Poisson–Dirichlet distribution. It suggests a simple algorithm of simulating the Kingman coalescent with occasional (simultaneous) multiple mergers.

Keywords

Ξ-coalescentexchangeabilityPoisson–Dirichlet distribution

MSC classification

Primary: 92D25: Population dynamics (general) 92D15: Problems related to evolution 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 839 - 854
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Billingsley, P. (1999). Weak Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Bolthausen, E., and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.Google Scholar
Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Adv. Appl. Prob. 6, 260–290.CrossRefGoogle Scholar
Donnelly, P., and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Prob. 27, 166–205.Google Scholar
Holst, L. (2001). The Poisson—Dirichlet distribution and its relatives revisited. Preprint, Department of Mathematics, Royal Institute of Technology, Stockholm. Available at http://www2.math.kth.se/~gunnare/lasse_e.htm.Google Scholar
Johnson, N. L., Kotz, S., and Balakrishnan, N. (1997). Discrete Multivariate Distributions. John Wiley, New York.Google Scholar
Kingman, J. F. C. (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, eds Koch, G. and Spizzichino, F., North-Holland, Amsterdam, pp. 97–112.Google Scholar
Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
Möhle, M. (1998). Robustness results for the coalescent. J. Appl. Prob. 35, 438–447.Google Scholar
Möhle, M., and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547–1562.Google Scholar
Möhle, M., and Sagitov, S. (2003). Coalescent patterns in exchangeable diploid population models. To appear in J. Math. Biol.Google Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.Google Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.Google Scholar
Schweinsberg, J. (2000). A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Commun. Prob. 5, No. 1.CrossRefGoogle Scholar
Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Prob. 5, No. 12.Google Scholar