关于随机环境下树指标马氏链的极限性质的研究

One important and popular research direction of the probability limit theory is the limit theory of stochastic structure indexed by a tree, an interdisciplinary subject, which has wide applications in the fields of computer algorithms, signal transmission etc. . In the project, we aim to study the limit theory of Markov chains indexed by a tree under the stochastic environment condition. According to the definition of Markov chains under the stochastic environment condition and Markov chains indexed by a tree, we propose the definition of Markov chains indexed by a tree under the stochastic environment condition and prove that this definition can be realized in probability Space. Meanwhile, the strong law of large numbers and the asymptotic equipartition property(AEP) of Markov chains indexed by a tree under the stochastic environment condition are also obtained. . Through the implementation of the project, we hope to solve the limit theory problem of Markov chains indexed by a tree under the stochastic environment condition. Furthermore, this method will deepen the connection of this field with the information theory.

树状随机结构的极限理论是概率极限理论的前沿热门研究方向，属于交叉学科，在计算机算法和无线信号传输等学科领域有着广泛的应用。. 本项目旨在研究随机环境下树指标马氏链的极限定理。通过随机环境下马氏链的定义以及树指标马氏链的定义，给出随机环境下树指标马氏链的定义，并且证明给出的定义能够在概率空间实现。在给出定义的同时，得到随机环境下树指标马氏链的强大数定律和渐进等分性。. 通过项目的实施，我们希望利用申请书中的方法不仅能够解决随机环境下树指标马氏链的极限理论问题，而且还可以进一步深化该领域与信息论的联系。

本项目旨在研究随机环境中树指标马氏链的强极限定理。通过随机环境下马氏链的定义以及树指标马氏链的定义，给出随机环境下树指标马氏链的定义，并且证明给出的定义能够在概率空间实现。在给出定义的同时，得到随机环境下树指标马氏链的强大数定律和渐近等分性。本项目中也研究了树指标随机过程的强偏差定理，高阶树指标马氏链的强大数定律及渐近等分性，以及非齐次马氏链的中心极限定理。 在本项目中，我们利用新的方法解决随机环境中树指标马氏链的极限理论问题，而且进一步深化该领域与信息论的联系。