Advertisement

Czechoslovak Mathematical Journal

, Volume 59, Issue 1, pp 23–37 | Cite as

A class of strong limit theorems for countable nonhomogeneous Markov chains on the generalized gambling system

  • Kangkang Wang
Article
  • 30 Downloads

Abstract

In this paper, we study the limit properties of countable nonhomogeneous Markov chains in the generalized gambling system by means of constructing compatible distributions and martingales. By allowing random selection functions to take values in arbitrary intervals, the concept of random selection is generalized. As corollaries, some strong limit theorems and the asymptotic equipartition property (AEP) theorems for countable nonhomogeneous Markov chains in the generalized gambling system are established. Some results obtained are extended.

Keywords

local convergence theorem stochastic adapted sequence martingale 

MSC 2000

60F15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Billingsley: Probability and Measure. Wiley, New York, 1986.zbMATHGoogle Scholar
  2. [2]
    R. V. Mises: Mathematical Theory of Probability and Statistics. Academic Press. New York, 1964.Google Scholar
  3. [3]
    A. N. Kolmogorov: On the logical foundation of probability theory. Lecture Notes in Mathematics. Springer-Verlag, New York, vol. 1021, 1982, pp. 1–2.Google Scholar
  4. [4]
    W. Liu and Z. Wang: An extension of a theorem on gambling systems to arbitrary binary random variables. Statistics and Probability Letters, vol. 28, 1996, pp. 51–58.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Z. Wang: A strong limit theorem on random selection for the N-valued random variables. Pure and Applied Mathematics 15 (1999), 56–61.zbMATHGoogle Scholar
  6. [6]
    W. Liu and W. Yang: An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains. Stochastic Process. Appl. 61 (1996), 279–292.Google Scholar
  7. [7]
    K. R. Stromberg and E. Hewitt: Real and abstract analysis-a modern treament of the theory of functions of real variable. Springer, New York, 1994.Google Scholar
  8. [8]
    C. Shannon: A mathematical theory of communication. Bell System Tech J. 27 (1948), 379–423.zbMATHMathSciNetGoogle Scholar
  9. [9]
    B. Mcmillan: The Basic Theorem of information theory. Ann. Math. Statist. 24 (1953), 196–219.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    L. Breiman: The individual ergodic theorem of information theory. Ann. Math. Statist. 28 (1957), 809–811.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    A. R. Barron: The strong ergodic theorem of densities; Generalized Shannon-McMillan-Breiman theorem. Ann. Probab. 13 (1985), 1292–1303.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    K. L. Chung: The ergodic theorem of information theorey. Ann. Math. Statist 32 (1961), 612–614.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. Feinstein: A new basic theory of information. IRE Trans. P.G.I.T. (1954), 2–22.Google Scholar
  14. [14]
    W. Yang and W. Liu: Strong law of large numbers and Shannon-McMillan theorem for Markov fields on trees. IEEE Trans. Inform. Theory 48 (2002), 313–318.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Z. Wang and W. Yang: Some strong limit theorems for both nonhomogeneous Markov chains of order two and their random transforms. J. Sys. Sci. and Math. Sci 24 (2004), 451–462.zbMATHGoogle Scholar
  16. [16]
    K. Wang and W. Yang: Research on strong limit theorem for Cantor-like stochastic sequence of Science and Technology (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 20 (2006), 26–29.zbMATHGoogle Scholar
  17. [17]
    K. Wang: Strong large number law for Markov chains field on arbitrary Cayley tree (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 20 (2006), 28–32.zbMATHGoogle Scholar
  18. [18]
    K. Wang: Some research on strong limit theorems for Cantor-like nonhomogeneous Markov chains (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 20 (2006), 19–23.zbMATHGoogle Scholar
  19. [19]
    K. Wang and Z. Qin: A class of strong limit theorems for arbitrary stochastic sequence in random selection system (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 20 (2006), 40–44.zbMATHMathSciNetGoogle Scholar
  20. [20]
    K. Wang: A class of strong limit theorems for stochastic sequence on product distribution in gambling system (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 21 (2007), 33–36.Google Scholar
  21. [21]
    K. Wang and H. Ye: A class of strong limit theorems for Markov chains field on arbitrary Bethe tree (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 21 (2007), 37–40.zbMATHGoogle Scholar
  22. [22]
    K. Wang: A class of strong limit theorems for random sum of Three-order countable nonhomogeneous Markov chains (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 21 (2007), 42–45.zbMATHGoogle Scholar
  23. [23]
    K. Wang and H. Ye: A class of local strong limit theorems for random sum of Cantor-like random function sequences (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 22 (2008), 87–90.zbMATHGoogle Scholar
  24. [24]
    K. Wang: A class of strong limit theorems on generalized gambling system for arbitrary continuous random variable sequence (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 22 (2008), 86–90.Google Scholar
  25. [25]
    M. Li: Some limit properties for the sequence of arbitrary random variables on their generalized random selection system (in Chinese). J. Jiangsu Univ. Sci-tech. Nat. Sci. 22 (2008), 90–94.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  • Kangkang Wang
    • 1
  1. 1.School of Mathematics and physicsJiangsu University of Science and TechnologyZhenjiangChina

Personalised recommendations