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.
Similar content being viewed by others
References
P. Billingsley: Probability and Measure. Wiley, New York, 1986.
R. V. Mises: Mathematical Theory of Probability and Statistics. Academic Press. New York, 1964.
A. N. Kolmogorov: On the logical foundation of probability theory. Lecture Notes in Mathematics. Springer-Verlag, New York, vol. 1021, 1982, pp. 1–2.
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.
Z. Wang: A strong limit theorem on random selection for the N-valued random variables. Pure and Applied Mathematics 15 (1999), 56–61.
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.
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.
C. Shannon: A mathematical theory of communication. Bell System Tech J. 27 (1948), 379–423.
B. Mcmillan: The Basic Theorem of information theory. Ann. Math. Statist. 24 (1953), 196–219.
L. Breiman: The individual ergodic theorem of information theory. Ann. Math. Statist. 28 (1957), 809–811.
A. R. Barron: The strong ergodic theorem of densities; Generalized Shannon-McMillan-Breiman theorem. Ann. Probab. 13 (1985), 1292–1303.
K. L. Chung: The ergodic theorem of information theorey. Ann. Math. Statist 32 (1961), 612–614.
A. Feinstein: A new basic theory of information. IRE Trans. P.G.I.T. (1954), 2–22.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wang, K. A class of strong limit theorems for countable nonhomogeneous Markov chains on the generalized gambling system. Czech Math J 59, 23–37 (2009). https://doi.org/10.1007/s10587-009-0002-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-009-0002-x