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Wandering phenomena in infinite-allelic diffusion models

Published online by Cambridge University Press:  01 July 2016

Tokuzo Shiga
Tokuzo Shiga*
Affiliation:
Tokyo Institute of Technology
*
∗Postal address: Department of Applied Physics, Tokyo Institute of Technology, Oh-Okayama, Tokyo, Japan.
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Abstract

We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

Keywords

WANDERING PHENOMENAWANDERING DISTRIBUTIONINFINITE-ALLELE MODELINFINITE-DIMENSIONAL DIFFUSION PROCESSESPOPULATION GENETICSOHTA–KIMURA MODELSTATIONARY DISTRIBUTIONEMPIRICAL MEAN PROCESSPROCESS OF CENTERED DISTRIBUTIONS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 3 , September 1982 , pp. 457 - 483
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Chung, K. L. (1960) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.CrossRefGoogle Scholar
[2] Dawson, D. A. and Hochberg, K. J. (1982) Wandering random measures in the Fleming-Viot model. To appear.CrossRefGoogle Scholar
[3] Ethier, S. N. (1981) A class of infinite dimensional distributions occurring in population genetics. Indiana Univ. Math. J. 30, 925–935.CrossRefGoogle Scholar
[4] Ethier, S. N. and Kurtz, T. G. (1981) The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.CrossRefGoogle Scholar
[5] Fleming, W. H. and Viot, M. (1979) Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843.CrossRefGoogle Scholar
[6] Ikeda, N. and Watanabe, S. (1981) Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam; Kodansha, Tokyo.Google Scholar
[7] Kingman, J. F. C. (1975) Coherent random walks arising in some genetic models. Proc. R. Soc. London A 351, 19–31.Google Scholar
[8] Moran, P. A. P. (1975) Wandering distributions and the electrophoretic profile. Theoret. Popn Biol. 8, 318–330.CrossRefGoogle ScholarPubMed
[9] Moran, P. A. P. (1976) Wandering distributions and the electrophoretic profile, II. Theoret. Popn Biol. 10, 145–149.Google Scholar
[10] Ohta, T. and Kimura, M. (1973) A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population. Genet. Res. Camb. 22, 202–204.Google Scholar
[11] Ohta, T. and Kimura, M. (1974) Simulation studies on electrophoretically detectable genetic variability in a finite population. Genetics 76, 614–624.Google Scholar
[12] Shiga, T. (1982) Multi-allelic stepping stone models in population genetics. J. Math. Kyoto Univ. 22, 1–40.Google Scholar
[13] Shiga, T. and Shimizu, A. (1980) Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20, 395–416.Google Scholar