Previous |  Up |  Next

Article

Title: A class of strong limit theorems for countable nonhomogeneous Markov chains on the generalized gambling system (English)
Author: Wang, Kangkang
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 59
Issue: 1
Year: 2009
Pages: 23-37
Summary lang: English
.
Category: math
.
Summary: 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. (English)
Keyword: local convergence theorem
Keyword: stochastic adapted sequence
Keyword: martingale
MSC: 60F15
MSC: 60G42
MSC: 60J10
idZBL: Zbl 1224.60055
idMR: MR2486613
.
Date available: 2010-07-20T14:49:26Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/140461
.
Reference: [1] Billingsley, P.: Probability and Measure.Wiley, New York (1986). Zbl 0649.60001, MR 0830424
Reference: [2] Mises, R. V.: Mathematical Theory of Probability and Statistics.Academic Press. New York (1964). Zbl 0132.12303, MR 0178486
Reference: [3] Kolmogorov, A. N.: On the logical foundation of probability theory. Lecture Notes in Mathematics.Springer-Verlag, New York, vol. 1021 (1982), 1-2. MR 0735967
Reference: [4] Liu, W., Wang, Z.: An extension of a theorem on gambling systems to arbitrary binary random variables.Statistics and Probability Letters, vol. 28 (1996), 51-58. MR 1394418, 10.1016/0167-7152(95)00081-X
Reference: [5] Wang, Z.: A strong limit theorem on random selection for the N-valued random variables.Pure and Applied Mathematics (1999), 15 56-61. MR 1762684
Reference: [6] Liu, W., Yang, W.: An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains.Stochastic Process. Appl. (1996), 61 279-292. Zbl 0861.60042, MR 1378852
Reference: [7] Stromberg, K. R., Hewitt, E.: Real and abstract analysis-a modern treament of the theory of functions of real variable.(1994), Springer, New York. MR 0367121
Reference: [8] Shannon, C.: A mathematical theory of communication.Bell System Tech J. (1948), 27 379-423. Zbl 1154.94303, MR 0026286, 10.1002/j.1538-7305.1948.tb01338.x
Reference: [9] Mcmillan, B.: The Basic Theorem of information theory.Ann. Math. Statist. (1953), 24 196-219. MR 0055621, 10.1214/aoms/1177729028
Reference: [10] Breiman, L.: The individual ergodic theorem of information theory.Ann. Math. Statist. (1957), 28 809-811. Zbl 0078.31801, MR 0092710, 10.1214/aoms/1177706899
Reference: [11] Barron, A. R.: The strong ergodic theorem of densities; Generalized Shannon-McMillan- Breiman theorem.Ann. Probab. (1985), 13 1292-1303. MR 0806226
Reference: [12] Chung, K. L.: The ergodic theorem of information theorey.Ann. Math. Statist (1961), 32 612-614. MR 0131782, 10.1214/aoms/1177705069
Reference: [13] Feinstein, A.: A new basic theory of information.IRE Trans. P.G.I.T. (1954), 2-22. MR 0088413
Reference: [14] Yang, W., Liu, W.: Strong law of large numbers and Shannon-McMillan theorem for Markov fields on trees.IEEE Trans. Inform. Theory (2002), 48 313-318. MR 1872187, 10.1109/18.971762
Reference: [15] Wang, Z., Yang, W.: Some strong limit theorems for both nonhomogeneous Markov chains of order two and their random transforms.J. Sys. Sci. and Math. Sci (2004), 24 451-462. MR 2108149
Reference: [16] Wang, K., Yang, W.: Research on strong limit theorem for Cantor-like stochastic sequence of Science and Technology (in Chinese).J. Jiangsu Univ. Sci-tech. Nat. Sci. (2006), 20 26-29.
Reference: [17] Wang, K.: Strong large number law for Markov chains field on arbitrary Cayley tree (in Chinese).J. Jiangsu Univ. Sci-tech. Nat. Sci. (2006), 20 28-32.
Reference: [18] Wang, K.: Some research on strong limit theorems for Cantor-like nonhomogeneous Markov chains (in Chinese).J. Jiangsu Univ. Sci-tech. Nat. Sci. (2006), 20 19-23.
Reference: [19] Wang, K., Qin, Z.: A class of strong limit theorems for arbitrary stochastic sequence in random selection system (in Chinese).J. Jiangsu Univ. Sci-tech. Nat. Sci. (2006), 20 40-44.
Reference: [20] Wang, K.: A class of strong limit theorems for stochastic sequence on product distribution in gambling system (in Chinese).J. Jiangsu Univ. Sci-tech. Nat. Sci. (2007), 21 33-36.
Reference: [21] Wang, K., Ye, H.: A class of strong limit theorems for Markov chains field on arbitrary Bethe tree (in Chinese).J. Jiangsu Univ. Sci-tech. Nat. Sci. (2007), 21 37-40.
Reference: [22] Wang, K.: 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. (2007), 21 42-45.
Reference: [23] Wang, K., Ye, H.: 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. (2008), 22 87-90.
Reference: [24] Wang, K.: 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. (2008), 22 86-90. MR 2445619
Reference: [25] Li, M.: 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. (2008), 22 90-94.
.

Files

Files Size Format View
CzechMathJ_59-2009-1_2.pdf 230.4Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo