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The latent roots of certain Markov chains arising in genetics: A new approach, I. Haploid models

Published online by Cambridge University Press:  01 July 2016

C. Cannings
C. Cannings*
Affiliation:
University of Sheffield
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Abstract

Haploid models of genetic drift in populations of constant size are considered. Generalizations of the models of Moran and Wright have been developed by Karlin and McGregor (for multiple alleles and non-overlapping generations), by Chia and Watterson (for two alleles and overlapping or non-overlapping generations) and by Chia (for multiple alleles and overlapping or non-overlapping generations), using conditioned branching processes. A new approach is developed which contains the models mentioned above and provides simpler expressions for the latent roots. A greater dependence between the birth events and death events can be permitted, and non-independent mutations treated.

Keywords

GENETIC DRIFTHAPLOIDLATENT ROOTSEXCHANGEABLE VECTORSNON-INDEPENDENT MUTATIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 2 , June 1974 , pp. 260 - 290
Copyright
Copyright © Applied Probability Trust 1974 

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References

Birkhoff, G. and MacLane, S. (1953) A Survey of Modern Algebra. MacMillan, New York.Google Scholar
Cannings, C. (1973) The equivalence of some overlapping and non-overlapping generation models for the study of genetic drift. J. Appl. Prob. 10, 432–436.CrossRefGoogle Scholar
Chia, A. B. (1968) Random mating in a population of cyclic size. J. Appl. Prob. 5, 21–30.CrossRefGoogle Scholar
Chia, A. B. (1969) Some finite stochastic models in genetics and their rates of evolution. , Monash University.Google Scholar
Chia, A. B. and Watterson, G. A. (1969) Demographic effects on the rate of genetic evolution. I. Constant size populations with two genotypes. J. Appl. Prob. 6, 231–249.CrossRefGoogle Scholar
Ewens, W. J. (1969) Population Genetics. Methuen, London.CrossRefGoogle Scholar
Feller, W. (1951) Diffusion processes in genetics. 2nd Berkeley Symposium on Mathematical Statistics and Probability..Google Scholar
Felsenstein, J. (1971) The rate of loss of multiple alleles in finite haploid populations. Theoret. Pop. Biol. 2, 391–403.Google Scholar
Gani, J. (1961) On the stochastic matrix in a genetic model of Moran. Biometrika 48, 203–206.Google Scholar
Karlin, S. (1968) Rates of approach to homozygosity for finite stochastic models with variable population size. Amer. Nat. 102, 443–455.Google Scholar
Karlin, S. and McGregor, J. (1962) Direct product branching processes and related Markov chains. Proc. Nat. Acad. Sci. USA 51, 598–602.Google Scholar
Karlin, S. and McGregor, J. (1965) Direct product branching processes and related induced Markoff chains. I. Calculation of rates of approach to homozygosity. Bernoulli, Bayes, Laplace Anniversary Volume. Springer-Verlag, Berlin. 11–145.Google Scholar
Karlin, S. and McGregor, J. (1968) The role of the Poisson progeny distribution in population genetic models. Math. Biosci. 2, 11–17.Google Scholar
Kempthorne, O. (1967) The concept of identity of genes by descent. 5th Berkeley Symp. Math. Statist. Prob..Google Scholar
Kimura, M. (1957) Some problems of stochastic processes in genetics. Ann. Math. Statist. 38, 882–901.Google Scholar
MacDuffee, C. C. (1946) The Theory of Matrices. Chelsea, New York.Google Scholar
Malécot, G. (1948) Les Mathématiques de L'hérédité. Masson, Paris.Google Scholar
Moran, P. A. P. (1958) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.Google Scholar
Moran, P. A. P. and Watterson, G. A. (1959) The genetic effects of family structure in natural populations. Austral. J. Biol. Sciences 12, 1–15.Google Scholar
Watterson, G. A. (1961) Markov chains with absorbing states, a genetic example. Ann. Math. Statist. 32, 716–729.CrossRefGoogle Scholar
Wright, S. W. (1931) Evolution in Mendelian populations. Genetics 16, 97–159.Google Scholar