Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME LIMIT PROPERTIES OF RANDOM TRANSITION PROBABILITY FOR SECOND-ORDER NONHOMOGENEOUS MARKOV CHAINS ON GENERALIZED GAMBLING SYSTEM INDEXED BY A DOUBLE ROOTED TREE

  • Wang, Kangkang (School of Mathematics and Physics, Jiangsu University of Science and Technology) ;
  • Zong, Decai (College of Computer Science and Engineering, Changshu Institute of Technology)
  • Received : 2010.12.30
  • Accepted : 2011.11.01
  • Published : 2012.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain on the generalized gambling system indexed by a tree by constructing a nonnegative martingale. As corollary, we obtain the property of the harmonic mean and the arithmetic mean of random transition probability for a second-order nonhomogeneous Markov chain indexed by a double root tree.

Keywords

References

  1. I. Benjamini and Y. Peres, Markov chains indexed by trees, Ann. Probab. 22(1994), 219-243. https://doi.org/10.1214/aop/1176988857
  2. T. Berger and Z. Ye, Entropic aspects of random fields on trees, IEEE Trans. Inform. Theory 36(1990), 1006-1018. https://doi.org/10.1109/18.57200
  3. H.L. Huang and W.G. Yang, Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree, Science in China 51(2008), 195-202.
  4. A.N. Kolmogorov, On the logical foundation of probability theory, Springer-Verlag. New York, 1982.
  5. W. Liu, A strong limit theorem for the harmonic mean of the random transition probabilities of finite nonhomogeneous Markov chains, Acta Mathematica Scientia, 20 (2000), 81-84. (In Chinese)
  6. W. Liu, A limit property of random conditional probabilities and an approach of conditional moment generating function, Acta Mathematicae Applicatae Sinica, 23 (2000), 275-279. (In Chinese)
  7. W. Liu, A limit property of random conditional probabilities, Statist. Probab. Lett., 49 (2000), 299-304 https://doi.org/10.1016/S0167-7152(00)00061-4
  8. R. Pemantle, Andomorphism invariant measure on tree, Ann. Probab, 20 (1992), 1549-1566 https://doi.org/10.1214/aop/1176989706
  9. Z.Y. Shi and W.G. Yang, Some limit properties of random transition probability for second- order nonhomogeneous Markov chains indexed by a tree, Journal of Inequalities and Applications, ID 503203. (2009)
  10. W.G. Yang, Some limit properties for Markov chains indexed by homogeneous tree , Stat. Probab. Letts. 65(2003), 241-250. https://doi.org/10.1016/j.spl.2003.04.001
  11. W.G. Yang and W. Liu, Strong law of large numbers for Markov chains fields on a bethe tree , Stat. Probab. Letts. 49(2000), 245-250. https://doi.org/10.1016/S0167-7152(00)00053-5
  12. W.G. Yang and Z. Ye, The asymptotic equipartition property for nonhomogeneous Markov chains indexed by a homogeneous tree, IEEE Trans. Inform. Theory 53(2007), 3275-3280 https://doi.org/10.1109/TIT.2007.903134
  13. Z. Ye and T. Berger, Ergodic regularity and asymptotic equipartition property of random fields on trees, Combin. Inform. System. Sci 21(1996), 157-184.
  14. Z. Ye and T. Berger, Information Measure for Discrete Random Fields, Science Press, Beijing, New York. 1998.