树上马氏链场的若干极限性质

Abstract

Abstract:

该文引进广义Ｂｅｔｈｅ树和广义Ｃａｙｌｅｙ树的概念，并研究其上马氏链场关于状态和状态序偶出现频率的强极限定理，作为主要结果的推论，得到Ｂｅｔｈｅ树和Ｃａｙｌｅｙ树上马氏链场的ＳｈａｎｎｏｎＭｃＭｉｌｌａｎ定理．证明中采用了研究概率论强极限定理的一种新的方法．

Key words: 树上的马氏链场, Ｂｅｔｈｅ树, Ｃａｙｌｅｙ树, 强极限定理, Ｓｈａｎｎｏｎ－ＭｃＭｉｌｌａｎ定理．

CLC Number:

60J10

LIU Wen, YANG Wei-Guo. Some Limit Properties for Markov Chain Fields on Trees[J].Acta mathematica scientia,Series A, 2001, 21(4): 512-520.

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１　ＳｐｉｔｚｅｒＦ．Ｍａｒｋｏｖｒａｎｄｏｍｆｉｅｌｄｓｏｎａｎｉｎｆｉｎｉｔｅｔｒｅｅ．Ａｎｎ．Ｐｒｏｂａｂ，１９７５，３：３８７－３９８２　ＢｅｒｇｅｒＴ，ＹｅＺ．Ｅｎｔｒｏｐｉｃａｓｐｅｃｔｓｏｆｒａｎｄｏｍｆｉｅｌｄｓｏｎｔｒｅｅｓ．ＩＥＥＥＴｒａｎｓｏｎｉｎｆｏｒｍａｔｉｏｎＴｈｅｏｒｙ，１９９０，３６（５）：１００６－１０１８
３　ＹｅＺ，ＢｅｒｇｅｒＴ．Ｅｒｇｏｄｉｃ，ｒｅｇｕｌａｒｔｙａｎｄａｓｙｍｐｔｏｔｉｃｅｑｕｉｐａｒｔｉｔｉｏｎｐｒｏｐｅｒｔｙｏｆｒａｎｄｏｍｆｉｅｌｄｓｏｎｔｒｅｅｓ．Ｊ．ｏｆＣｏｍｂｉｎａｔｏｒｉｃｓ，Ｉｎｆｏｒｍａｔｉｏｎ＆ＳｙｓｔｅｍＳｃｉｅｎｃｅｓ，１９９６，２１（２）：１５７－１８４
４　ＹｅＺ，ＢｅｒｇｅｒＴ．ＩｎｆｏｒｍａｔｉｏｎＭｅａｓｕｒｅｆｏｒＤｉｓｃｒｅｔｅＲａｎｄｏｍＦｉｅｌｄｓ．Ｂｅｉｊｉｎｇ，ＮｅｗＹｏｒｋ：ＳｃｉｅｎｃｅＰｒｅｓｓ，１９９８
５　ＢｅｎｊａｍｉｎｉＩ，ＰｅｒｅｓＹ．Ｍａｒｋｏｖｃｈａｉｎｓｉｎｄｅｘｅｄｂｙｔｒｅｅｓ．Ａｎｎ．Ｐｒｏｂａｂ，１９９４，２２：２１９－２４３
６　ＦｅｌｌｅｒＷ．ＡｎＩｎｔｒｏｄｕｃｔｉｏｎｔｏＰｒｏｂａｂｉｌｉｔｙＴｈｅｏｒｙａｎｄＩｔｓＡｐｐｌｉｃａｔｉｏｎｓ．Ｖｏｌ．１，２ｎｄｅｄ．ＷｉｌｅｙＮｅｗＹｏｒｋ：１９５７
７　ＬｉｕＷ，ＹａｎｇＷ．Ｇ．ＡｌｉｍｉｔｔｈｅｏｒｅｍｆｏｒｔｈｅｅｎｔｒｏｐｙｄｅｎｓｉｔｙｏｆｎｏｎｈｏｍｏｇｅｎｅｏｕｓＭａｒｋｏｖｉｎｆｏｒｍａｔｉｏｎｓｏｕｒｃｅ．ＳｔａｔｉｓＰｒｏｂａｂＬｅｔｔｅｒｓ，１９９５，２２：２９５－３０１
８　ＬｉｕＷ，ＹａｎｇＷ．Ｇ．ＡｎｅｘｔｅｎｓｉｏｎｏｆＳｈａｎｎｏｎ－ＭｃＭｉｌｌａｎｔｈｅｏｒｅｍａｎｄｓｏｍｅｌｉｍｉｔｐｒｏｐｅｒｔｉｅｓｆｏｒｎｏｎｈｏｍｏｇｅｎｅｏｕｓＭａｒｋｏｖｃｈａｉｎｓ．ＳｔｏｃｈａｓｔｉｃＰｒｏｃｅｓｓ．Ａｐｐｌ．１９９６，６１：１２９－１４６

[1]	LI Ying-Qiu, WANG He-Song, WANG Zhong. Some Strong Limit Theorems of Geometric Average of Transition Probabilities \|and Their Function for Markov Chains in Markovian Environments [J]. Acta mathematica scientia,Series A, 2011, 31(2): 508-517.
[2]	WANG Shi-Dong, HONG Wen-Ming. Alternative Proof for the Recurrence and Transience of Random Walks in Random Environment with Bounded Jumps [J]. Acta mathematica scientia,Series A, 2010, 30(2): 289-296.
[3]	LIU Xiang-Dong, DAI Yong-Long. A Class of Random Walks on Half line in Random Environments [J]. Acta mathematica scientia,Series A, 2005, 25(1): 98-102.

Some Limit Properties for Markov Chain Fields on Trees

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