非齐次马氏链广义熵遍历定理的推广

发布时间：2018-10-05 09:30

【摘要】：马尔可夫链是概率论研究中的一类重要的随机过程,在计算科学、随机分形、经济学、医学、工业学等社会科学中有着广泛的应用。近年来,汪忠志和杨卫国在给出非齐次马氏链的广义熵密度之后,得到了关于非齐次马氏链的一类极限定理即广义熵遍历定理。本文将运用鞅方法将广义熵遍历定理分别推广到一阶非齐次马氏链的一类二元函数上和二阶非齐次马氏信源上。首先,本文简略介绍了马尔可夫过程的一些研究背景和国内外的主要研究成就以及本文的结构安排。随后,本文给出了马氏链和鞅论中的一些基础知识以及马氏链的研究过程中一些比较重要的引理。然后,本文首先介绍了非齐次马氏链广义熵密度的定义,以及杨卫国得到的非齐次马氏链的广义熵遍历定理。接着本文运用鞅方法把一阶非齐次马氏链的极限定理推广到一类函数上。此外,在生活中,我们往往要用二阶马氏信源去描述实际问题。杨卫国和刘文已经得到了关于二阶非齐次马氏信源的经典熵遍历定理。为此,本文在给出二阶非齐次马氏信源的广义熵密度的前提下,把广义熵遍历定理推广到二阶非齐次马氏信源上。并得到二阶非齐次马氏信源的一类极限定理即二阶非齐次马氏信源的广义熵遍历定理。最后,对全文进行了总结,阐述了本文中的一些不足,并表明了以后的研究内容与探索方向。
[Abstract]:Markov chain is an important stochastic process in the study of probability theory. It has been widely used in computational science, random fractal, economics, medicine, industry and other social sciences. In recent years, after giving the generalized entropy density of nonhomogeneous Markov chains, Wang Zhongzhi and Yang Weiguo have obtained a class of limit theorems about nonhomogeneous Markov chains, that is, generalized entropy ergodic theorems. In this paper, the generalized entropy ergodic theorem is extended to a class of binary functions of first order nonhomogeneous Markov chains and second-order nonhomogeneous Markov information sources by means of martingale method. First of all, this paper briefly introduces the background of Markov process, the main research achievements at home and abroad, and the structure of this paper. Then, some basic knowledge of Markov chain and martingale theory and some important Lemma in the study of Markov chain are given. Then, the definition of generalized entropy density of nonhomogeneous Markov chains and the generalized entropy ergodic theorem of nonhomogeneous Markov chains obtained by Yang Weiguo are introduced. Then we generalize the limit theorem of first order nonhomogeneous Markov chain to a class of functions by using martingale method. In addition, in life, we often use a second-order Markov source to describe practical problems. Yang Weiguo and Liu Wen have obtained the classical entropy ergodic theorem on the second order nonhomogeneous Markov sources. In this paper, we generalize the generalized entropy ergodic theorem to the second order inhomogeneous Markov information source on the premise of giving the generalized entropy density of the second order nonhomogeneous Markov information source. The generalized entropy ergodic theorem of second-order nonhomogeneous Markov sources is obtained. Finally, the paper summarizes the full text, expounds some shortcomings in this paper, and points out the research content and exploration direction in the future.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224

【参考文献】