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The general coalescent with asynchronous mergers of ancestral lines

Published online by Cambridge University Press:  14 July 2016

Serik Sagitov
Serik Sagitov*
Affiliation:
Chalmers University of Technology and Göteborg University
*
∗Postal address: Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden. Email address: [email protected].
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Abstract

Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral process. We describe a class of asymptotic structures for the ancestral process via a convergence criterion. One of the basic conditions of the criterion prevents simultaneous mergers of ancestral lines. Another key condition implies that the marginal distribution of the family size is attracted by an infinitely divisible distribution. If the latter is normal the coalescent allows only for pairwise mergers (Kingman's coalescent). Otherwise multiple mergers happen with positive probability.

Keywords

Coalescentinfinitely divisible distributionde Finetti's theoremreduced branching process

MSC classification

Primary: 92D25: Population dynamics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1116 - 1125
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This work is part of the Bank of Sweden Tercentenary Foundation project ‘Dependence and Interaction in Stochastic Population Dynamics’.

References

Feller, W. (1966). An Introduction to Probability Theory and its Applications, Vol 2. John Wiley, New York.Google Scholar
Fleischmann, K., and Siegmund-Shultze, R. (1977). The structure of reduced Galton–Watson processes. Math. Nachr. 79, 233–241.CrossRefGoogle Scholar
Kingman, J. F. C. (1982a). On the genealogy of large populations. In Essays in Statistical Science, eds. Gani, J. and Hannan, E. J. (J. Appl. Prob. 19A), Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
Kingman, J. F. C. (1982b). 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. (1982c). The coalescent. Stoch. Proc. Appl. 13, 235–248.CrossRefGoogle Scholar
Möhle, M. (1998). Robustness results for the coalescent. J. Appl. Prob. 35, 438–447.CrossRefGoogle Scholar
Slack, R. S. (1968). A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.CrossRefGoogle Scholar
Yakymiv, A. L. (1980). Reduced branching processes. Theor. Prob. Appl. 25, 584–588.CrossRefGoogle Scholar
Zolotarev, V. M. (1957). More exact statements of several theorems in the theory of branching processes. Teor. Veroyatnost. i Primemen. 2, 256–266. (In Russian.)Google Scholar