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The coalescent process in a population with stochastically varying size

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ingemar Kaj  and
Stephen M. Krone
Ingemar Kaj*
Affiliation:
Uppsala University
Stephen M. Krone*
Affiliation:
University of Idaho
*
∗ Postal address: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden.
∗∗ Postal address: Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, USA. Email address: krone@uidaho.edu
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Abstract

We study the genealogical structure of a population with stochastically fluctuating size. If such fluctuations, after suitable rescaling, can be approximated by a nice continuous-time process, we prove weak convergence in the Skorokhod topology of the scaled ancestral process to a stochastic time change of Kingman's coalescent, the time change being given by an additive functional of the limiting backward size process.

Keywords

Coalescentstochastically fluctuating population sizeweak convergencestochastic time change

MSC classification

Primary: 92D10: Genetics 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60F05: Central limit and other weak theorems
Secondary: 92D25: Population dynamics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 33 - 48
Copyright
Copyright © Applied Probability Trust 2003 

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References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Donnelly, P. (1986). A genealogical approach to variable-population-size models in population genetics. J. Appl. Prob. 23, 283–296.Google Scholar
Donnelly, P., and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Prob. 27, 166–205.CrossRefGoogle Scholar
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994). Sampling theory for neutral alleles in a varying environment. Phil. Trans. R. Soc. London B 344, 403–410.Google Scholar
Kaj, I., Krone, S. M., and Lascoux, M. (2001). Coalescent theory for seed bank models. J. Appl. Prob. 38, 285–300.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Möhle, M. (2000). Ancestral processes in population genetics—the coalescent. J. Theoret. Biol. 204, 629–638.Google Scholar
Möhle, M. (2002). The coalescent in population models with time-inhomogeneous environment. Stoch. Process. Appl. 97, 199–227.Google Scholar
Nordborg, M., and Krone, S. M. (2002). Separation of time scales and convergence to the coalescent in structured populations. In Modern Developments in Theoretical Population Genetics, eds Slatkin, M. and Veuille, M., Cambridge University Press, pp. 194–232.Google Scholar
Rogers, L. C. G., and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. 2. John Wiley, Chichester.Google Scholar
Tavaré, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetic models. Theoret. Pop. Biol. 26, 119–164.Google Scholar