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An ergodic theorem for evolution in a random environment

Published online by Cambridge University Press:  14 July 2016

M. Frank Norman
M. Frank Norman*
Affiliation:
University of Pennsylvania
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Abstract

Let w1, w12 and w2 be the fitnesses of genotypes A1A1, A1A2 and A2A2 in an infinite diploid population, and let pn be the A1, gene frequency in the nth generation. If fitness varies independently from generation to generation, then pn is a Markov process with a continuum of states. If E[In(wi/w12)] < 0 for i = 1, 2, then there is a unique stationary probability, and the distribution of pn converges to it as n → ∞.

Keywords

GENETIC MODELSRANDOM ENVIRONMENTSMARKOV PROCESSESERGODIC THEORY
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 4 , December 1975 , pp. 661 - 672
Copyright
Copyright © Applied Probability Trust 1975 

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References

Breiman, L. (1968) Probability. Addison-Wesley, Reading, Massachusetts.Google Scholar
Chung, K. L. (1974) A Course in Probability Theory, 2nd ed. Academic Press, New York.Google Scholar
Ewens, W. J. (1969) Population Genetics. Methuen, London.CrossRefGoogle Scholar
Gillespie, J. (1973) Polymorphism in random environments. Theor. Pop. Biol. 4, 193–195.Google Scholar
Karlin, S. and Lieberman, U. (1974) Random temporal variation in selection intensities: case of large population size. Theor. Pop. Biol. 6, 355–382.Google Scholar
Lamperti, J. (1960) Criteria for recurrence or transience of stochastic processes, I. J. Math. Anal. Appl. 1, 314–330.Google Scholar