关于随机环境中树指标马氏链的定义及存在性
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摘要
给出了离散状态下随机环境中树指标马氏链的定义,它是树指标马氏链概念的推广,也是随机环境中马氏链概念的推广.文章同时研究其若干等价定理,证明了随机环境中树指标马氏链的存在性,以及马氏环境中树指标马氏链与树指标马氏双链的等价性.
This paper gave the definition of tree-indexed Markov chains in random environment under the discrete state,which generalized both the concept of treeindexed Markov chains and the concept of Markov chains in random environment.Meanwhile,we studied some equivalent theorems of tree-indexed Markov chains in random environment and proved the existence about it.At last,we gave the proof of equivalence between tree-indexed Markov chains in Markov environment and treeindexed Markov double chains.
This paper gave the definition of tree-indexed Markov chains in random environment under the discrete state,which generalized both the concept of treeindexed Markov chains and the concept of Markov chains in random environment.Meanwhile,we studied some equivalent theorems of tree-indexed Markov chains in random environment and proved the existence about it.At last,we gave the proof of equivalence between tree-indexed Markov chains in Markov environment and treeindexed Markov double chains.
引文
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[2]Chen X X,Yang W G,Wang B.The equivalent definition of T-indexed Markov chains.Journal of Mathematical Study,2012,45(4):411-414.
[3]Berger T,Ye Z.Entropic aspects of random fields on trees.IEEE Thins.Inform.Theory,1990,36(5):1006-1018.
[4]Ye Z,Berger T.Ergodic,regulary and asymptotic equipartition property of random fields on trees.J.Combin.Inform.System.Sci.,1996,21:1570-184.
[5]Ye Z,Berger T.Information Measures for Discrete Random Fields.Beijing:Science,1998.
[6]Pemantle R.Antomorphism invariant measure on trees.Ann.Probab.,1992,20:1549-1566.
[7]Yang W G,Liu W.Strong law of large numbers for Markov chains fields on a Bethe tree.Statist.Probab.Lett,2000,49(3):245-250.
[8]Yang W G.Some limit properties for Markov chains indexed by a homogeneous tree.Statist.Probab.Lett,2003,65(3):241-250.
[9]Yang W G,Ye Z.The asymptotic equipartition property for Markov chains indexed by a Homogeneous tree.IEEE Trans.Inf.Theory,2007,53(9):3275-3280.
[10]Huang H L,Yang W G.Strong law of large number for Markov chains indexed by an infinite tree with uniformly bounded degree.Science in China,2008,51(2):195-202.
[11]Shi Z Y,Yang W G.A limit property of random transition probabihty for a nonhomogeneous Markov chains indexed by a tree.Acta Mathematicae Applicatae Since,2008,31(4):648-653(in Chinese).
[12]Wang B,Yang W G,Shi Z Y.Strong laws of large numbers for countable Markov chains indexed by a Cayley tree.Science in China,2012,42(10):1031-1036.
[13]Nawrotzki K.Discrete open system or Markov chains in a random environment,I.J.Inform.Process,Cybernet,1981,17:569-599.
[14]Nawrotzki K.Discrete open system or Markov chains in a random environment,II.J.Inform.Process,Cybernet,1982,18:83-98.
[15]Cogburn R.On the central limit theorem for Markov chains in random environments.Ann.Probab.,1991,19(2):587-604.
[16]Cogburn R.The ergodic theory of Markov chains in random environments.Z.Wahrsch Verw.Gebiete.,1984,66(1):109-128.
[17]Cogburn R.On direct convergence and periodicity for transition probabilities of Markov chains in random environments.Ann.Probab.,1990,18(2):642-654.
[18]Hu D H.The construction of Markov processes in random environments and the equivalence theorems.Science in China(Series A),2004,47(4):481-496.
[19]Hu D H.The existence and uniqueness of q-processes in random environments.Science in China(Series A),2004,47(5):641-658.
[20]李应求.双无限环境中马氏链的瞬时性与不变函数.数学年刊,2003,24(2):515-520.
[21]李应求,王苏明,胡杨利.马氏环境中马氏链的一类强极限定理.数学进展,2008,37(5):543-550.
[22]Guyon J.Limit theorems for bifurcating Markov chains—Application to the detection of cellular aging.The Annals,of Applied Probability,2007,17(5/6):1538-1569.
[23]严加安.测度论讲义.北京:科学出版社,2004.
[2]Chen X X,Yang W G,Wang B.The equivalent definition of T-indexed Markov chains.Journal of Mathematical Study,2012,45(4):411-414.
[3]Berger T,Ye Z.Entropic aspects of random fields on trees.IEEE Thins.Inform.Theory,1990,36(5):1006-1018.
[4]Ye Z,Berger T.Ergodic,regulary and asymptotic equipartition property of random fields on trees.J.Combin.Inform.System.Sci.,1996,21:1570-184.
[5]Ye Z,Berger T.Information Measures for Discrete Random Fields.Beijing:Science,1998.
[6]Pemantle R.Antomorphism invariant measure on trees.Ann.Probab.,1992,20:1549-1566.
[7]Yang W G,Liu W.Strong law of large numbers for Markov chains fields on a Bethe tree.Statist.Probab.Lett,2000,49(3):245-250.
[8]Yang W G.Some limit properties for Markov chains indexed by a homogeneous tree.Statist.Probab.Lett,2003,65(3):241-250.
[9]Yang W G,Ye Z.The asymptotic equipartition property for Markov chains indexed by a Homogeneous tree.IEEE Trans.Inf.Theory,2007,53(9):3275-3280.
[10]Huang H L,Yang W G.Strong law of large number for Markov chains indexed by an infinite tree with uniformly bounded degree.Science in China,2008,51(2):195-202.
[11]Shi Z Y,Yang W G.A limit property of random transition probabihty for a nonhomogeneous Markov chains indexed by a tree.Acta Mathematicae Applicatae Since,2008,31(4):648-653(in Chinese).
[12]Wang B,Yang W G,Shi Z Y.Strong laws of large numbers for countable Markov chains indexed by a Cayley tree.Science in China,2012,42(10):1031-1036.
[13]Nawrotzki K.Discrete open system or Markov chains in a random environment,I.J.Inform.Process,Cybernet,1981,17:569-599.
[14]Nawrotzki K.Discrete open system or Markov chains in a random environment,II.J.Inform.Process,Cybernet,1982,18:83-98.
[15]Cogburn R.On the central limit theorem for Markov chains in random environments.Ann.Probab.,1991,19(2):587-604.
[16]Cogburn R.The ergodic theory of Markov chains in random environments.Z.Wahrsch Verw.Gebiete.,1984,66(1):109-128.
[17]Cogburn R.On direct convergence and periodicity for transition probabilities of Markov chains in random environments.Ann.Probab.,1990,18(2):642-654.
[18]Hu D H.The construction of Markov processes in random environments and the equivalence theorems.Science in China(Series A),2004,47(4):481-496.
[19]Hu D H.The existence and uniqueness of q-processes in random environments.Science in China(Series A),2004,47(5):641-658.
[20]李应求.双无限环境中马氏链的瞬时性与不变函数.数学年刊,2003,24(2):515-520.
[21]李应求,王苏明,胡杨利.马氏环境中马氏链的一类强极限定理.数学进展,2008,37(5):543-550.
[22]Guyon J.Limit theorems for bifurcating Markov chains—Application to the detection of cellular aging.The Annals,of Applied Probability,2007,17(5/6):1538-1569.
[23]严加安.测度论讲义.北京:科学出版社,2004.