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An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains

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  • Wen, Liu
  • Weiguo, Yang

Abstract

Let {Xn, n >= 0} be a Markov chains with the state space S = {1, 2, ..., m}, and the probability distribution P(x0) [Pi]nk=1Pk(xkxk-1), where Pk(ji) is the transition probability P(Xk = jXk-1 = i). Let gk(i, j) be the functions defined on S x S, and let Fn([omega]) = (1/n)[Sigma]nk=1gk(Xk-1, Xk). In this paper the limit properties of Fn([omega]) and the relative entropy density fn([omega]) = -(1/n)[logP(X0) + [Sigma]nk=1logPk(XkXk-1] are studied, and some theorems on a.e. convergence for {Xn, n >= 0} are obtained, and the Shannon-McMillan theorem is extended to the case of nonhomogeneous Markov chains.

Suggested Citation

  • Wen, Liu & Weiguo, Yang, 1996. "An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 61(1), pages 129-145, January.
    • Handle: RePEc:eee:spapps:v:61:y:1996:i:1:p:129-145
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    Cited by:

    1. Liu, Wen, 2003. "Some limit properties of the multivariate function sequences of discrete random variables," Statistics & Probability Letters, Elsevier, vol. 61(1), pages 41-50, January.
    2. Liu, Wen, 2000. "A limit property of random conditional probabilities," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 299-304, September.

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