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Entropy and Martingales in Markov Chain Models

E. Seneta
Journal of Applied Probability
Vol. 19, Essays in Statistical Science (1982), pp. 367-381
DOI: 10.2307/3213576
Stable URL: http://www.jstor.org/stable/3213576
Page Count: 15
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Entropy and Martingales in Markov Chain Models
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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 E(Xk). The model is used for a brief probabilistic assessment of the relationship between Boltzmann's H-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 E(Xk).

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