随机环境中树指标随机场的极限定理

项目名称： 随机环境中树指标随机场的极限定理

项目编号： No.11201344

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 黄辉林

作者单位： 温州大学

项目金额： 22万元

中文摘要： 树指标随机场是一个非常新的研究方向，它已引起物理学、 概率论及信息论界的广泛兴趣。本项目拟对随机环境中树指标马氏链场的极限定理展开深入研究。首先，利用鞅收敛定理建立有限状态空间马氏环境中树指标马氏链的强大数定律和几乎处处收敛意义下的熵定理(又称渐近等分性)。其次，拟研究有限状态空间马氏环境中树指标马氏链的转移概率调和平均的强极限定理。最后，通过引入渐近对数似然比作为有限状态空间随机环境中齐次树指标任意随机场马尔科夫逼近的一种度量，再利用鞅方法建立一类随机环境中齐次树指标任意随机场泛函的强偏差定理。本项目的研究将为我们进一步完善树指标随机过程的极限理论奠定良好的基础，也可为我们继续研究其他网络上的随机过程提供研究思路。

中文关键词： 树指标马氏链；随机环境；强大数定律；渐近等分性；树指标循环马氏链

英文摘要： The subject of tree-indexed random fields is very young. The tree model has recently drawn increasing interest from specialists in physics, probability and information theory. This project is in depth study on limit theorems for Markov chains indexed by trees in random environment. Firstly, we plan to establishthe strong laws of large numbers and the asymptotic equipartition property for Markov chains indexed by trees in Markovian environment with finit state spaces by using the Doob martingale convergence theorem. Secondly，a strong limit theorem will be established, which is for the harmonic mean of the transition probabilities for Markov chains indexed by trees in random enviroment with finit state spaces. Finally, by introducing the asymptotic logarithmic likelihood ratio as a measure of Markov approximation of the arbitrary random fields on a homogeneous tree in random enviroment, we estabilish a class of small deviation theorems for functionals of random fields indexed by trees in random enviroment with finit state spaces. The study of the project, which can also provide perspectives to continue to study the stochastic processes on other networks, will lay a good foundation for further improving on the limit theory of stochastic processes indexed by trees.

英文关键词： Markov chains indexed by trees；random environments；strong law of large numbers；AEP；circular markov chain indexed by trees