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Asymptotic Bahavior of the Moran Particle System

Part of: Limit theorems

Published online by Cambridge University Press:  22 February 2016

Shui Feng  and
Jie Xiong
Shui Feng*
Affiliation:
McMaster University
Jie Xiong*
Affiliation:
University of Macau and University of Tennessee
*
∗ Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton Hall #211, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada. Email address: [email protected]
∗∗ Postal address: Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira, Taipa, Macau, China.
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Abstract

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The asymptotic behavior is studied for an interacting particle system that involves independent motion and random sampling. For a fixed sampling rate, the empirical process of the particle system converges to the Fleming-Viot process when the number of particles approaches ∞. If the sampling rate approaches 0 as the number of particles becomes large, the corresponding empirical process will converge to the deterministic flow of the motion. In the main results of this paper, we study the corresponding central limit theorems and large deviations. Both the Gaussian limits and the large deviations depend on the sampling scales explicitly.

Keywords

Exponential tightnessMoran particle processFleming-Viot processgenealogylarge deviationsparticle representation

MSC classification

Primary: 60F10: Large deviations
Secondary: 92D10: Genetics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 2 , June 2013 , pp. 379 - 397
Copyright
© Applied Probability Trust 

Footnotes

Research supported by the Natural Science and Engineering Research Council of Canada.

Research partially supported by NSF DMS-0906907.

References

Asselah, A., Ferrari, P. A. and Groisman, P. (2011). Quasistationary distributions and Fleming–Viot processes in finite spaces. J. Appl. Prob. 48, 322–332.Google Scholar
Burdzy, K., Holyst, R. and March, P. (2000). A Fleming-Viot particle representation of the Dirichlet Laplacian. Commun. Math. Phys. 214, 679–703.Google Scholar
Dawson, D. A. (1993). Measure-valued Markov processes. In Ecole d'Été de Probabilités de Saint Flour XXI (Lecture Notes Math. 1541), Springer, Berlin, pp. 1–260.CrossRefGoogle Scholar
Dawson, D. A. and Feng, S. (1998). Large deviations for the Fleming-Viot process with neutral mutation and selection. Stoch. Process. Appl. 77, 207–232.Google Scholar
Dawson, D. A. and Feng, S. (2001). Large deviations for the Fleming-Viot process with neutral mutation and selection. II. Stoch. Process. Appl. 92, 131–162.Google Scholar
Dawson, D. A. and Gärtner, J. (1987). Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20, 247–308.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, New York.CrossRefGoogle Scholar
Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Prob. 24, 698–742.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1993). Fleming–Viot processes in population genetics. SIAM J. Control Optimization 31, 345–386.Google Scholar
Feng, S. and Xiong, J. (2002). Large deviations and quasi-potential of a Fleming-Viot process. Electron. Commun. Prob. 7, 13–25.Google Scholar
Fleming, W. H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843.Google Scholar
Grigorescu, I. (2007). Large deviations for a catalytic Fleming-Viot branching system. Commun. Pure Appl. Math. 60, 1056–1080.Google Scholar
Kallianpur, G. and Xiong, J. (1995). Stochastic Differential Equations on Infinite-Dimensional Spaces (IMS Lecture Notes Monogr. Ser. 26). Institute of Mathematical Statistics, Hayward, CA.CrossRefGoogle Scholar
Mitoma, I. (1983). Tightness of probabilities on C([0,1];S') and D([0,1];S'). Ann. Prob. 11, 989–999.Google Scholar
Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.Google Scholar
Puhalskii, A. (1991). On functional principle of large deviations. In New Trends in Probability and Statistics, Vol. 1, eds Sazonov, V. and Shervashidze, T., VSP, Utrecht, pp. 198–218.Google Scholar
Xiang, K.-N. and Zhang, T.-S. (2005). Small time asymptotics for Fleming-Viot processes. Infin. Dimens. Anal. Quantum Prob. Relat. Top. 8, 605–630.CrossRefGoogle Scholar