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Entropy and martingales in Markov chain models

Published online by Cambridge University Press:  14 July 2016

E. Seneta
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Abstract

The concept of entropy in models is discussed with particular reference to the work of P.A.P. Moran. For a vector-valued Markov chain {Xk} whose states are relative-frequency (proportion) tables corresponding to a physical mixing model of a number N of particles over n urns, the definition of entropy may be based on the usual information-theoretic concept applied to the probability distribution given by the expectation . The model is used for a brief probabilistic assessment of the relationship between Boltzmann's Η-Theorem, the Ehrenfest urn model, and Poincaré's considerations on the mixing of liquids and card shuffling, centred on the property of an ultimately uniform distribution of a single particle. It is then generalized to the situation where the total number of particles fluctuates over time, and martingale results are used to establish convergence for .

Keywords

ENTROPYMARKOV CHAINSURN MODELSMARTINGALESFLUCTUATING POPULATION SIZE
Type
Part 6 — Stochastic Processes
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 367 - 381
Copyright
Copyright © 1982 Applied Probability Trust 

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References

[1] Ash, R.B. (1972) Real Analysis and Probability. Academic Press, New York.Google Scholar
[2] Bernoulli, D. (1769) Disquisitiones analyticae de novo problemate conjecturale. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 14, pars 1, 3–25.Google Scholar
[3] Ehrenfest, P. and Ehrenfest, T. (1907) Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem. Phys. Z. 8, 311–314.Google Scholar
[4] Ewens, W.J. (1969) Population Genetics. Methuen, London.Google Scholar
[5] Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[6] Fisher, R.A. (1958) The Genetical Theory of Natural Selection , 2nd edn. Dover, New York.Google Scholar
[7] Johnson, N.L. and Kotz, S. (1977) Urn Models and their Application. Wiley, New York.Google Scholar
[8] Kac, M. (1947) Random walk and the theory of Brownian motion. Amer. Math. Monthly 54, 369–391. Reprinted (1954); Selected Papers on Noise and Stochastic Processes , ed. Wax, N., Dover, New York, 295–317.Google Scholar
[9] Kac, M. (1959) Probability and Related Topics in Physical Sciences. Interscience, London.Google Scholar
[10] Kac, M. (1964) Probability. Scientific American 211 (September), 92–108.CrossRefGoogle Scholar
[11] Khinchin, A.I. (1957) Mathematical Foundations of Information Theory. Dover, New York.Google Scholar
[12] Kohlrausch, K.W.F. and Schrödinger, ?. (1926) Das Ehrenfestsche Modell der ?-Kurve. Phys. Z. 27, 306–313.Google Scholar
[13] Li, C.C. (1967) Genetic equilibrium under selection. Biometrics 23, 397–484.CrossRefGoogle ScholarPubMed
[14] Moran, P.A.P. (1959–1960) The survival of a mutant under selection, I and II. J. Austral. Math. Soc. 1, 121–126, 485–491.Google Scholar
[15] Moran, P.A.P. (1961) Entropy, Markov processes and Boltzmann's ?-Theorem. Proc. Camb. Phil. Soc. 57, 833–842.Google Scholar
[16] Moran, P.A.P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
[17] Moran, P.A.P. (1964) On the non-existence of adaptive topographies. Ann. Human Genet. 27, 383–393.CrossRefGoogle Scholar
[18] Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
[19] Poincaré, H. (1912) Calcul des Probabilités , deuxième édition. Gauthier–Villars, Paris.Google Scholar
[20] Seneta, ?. (1978) A relaxation view of a genetic problem. Adv. Appl. Prob. 10, 716–720.Google Scholar
[21] Sheynin, O.B. (1972) D. Bernoulli's work on probability. Rete 1, 273–299.Google Scholar
[22] Urban, F.M. (1932) Das Mischungsproblem des Daniel Bernoulli. Atti Del Congresso Internazionale Dei Matematici, Bologna , 1928, 6, 21–25.Google Scholar