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The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems

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  • Liu, Wen
  • Yang, Weiguo

Abstract

Let {Xn, n[greater-or-equal, slanted]0} be a sequence of random variables on the probability space ([Omega],F,P) taking values in the alphabet S={1,2,...,N}, and Q be another probability measure on F, under which {Xn, n[greater-or-equal, slanted]0} is a Markov chain. Let h(P Q) be the sample divergence rate of P with respect to Q related to {Xn}. In this paper the Markov approximation of {Xn, n[greater-or-equal, slanted]0} under P is discussed by using the notion of h(P Q), and a class of small deviation theorems for the averages of the bivariate functions of {Xn, n[greater-or-equal, slanted]0} are obtained. In the proof an analytic technique in the study of a.e. convergence together with the martingale convergence theorem is applied.

Suggested Citation

  • Liu, Wen & Yang, Weiguo, 2000. "The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 117-130, September.
    • Handle: RePEc:eee:spapps:v:89:y:2000:i:1:p:117-130
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    Cited by:

    1. Yang, Weiguo, 2003. "Some limit properties for Markov chains indexed by a homogeneous tree," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 241-250, November.
    2. Liu, Wen & Wang, Liying, 2003. "The Markov approximation of the random fields on Cayley trees and a class of small deviation theorems," Statistics & Probability Letters, Elsevier, vol. 63(2), pages 113-121, June.
    3. Dong, Yan & Yang, Weiguo & Bai, Jianfang, 2011. "The strong law of large numbers and the Shannon–McMillan theorem for nonhomogeneous Markov chains indexed by a Cayley tree," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1883-1890.

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