关于非齐次马氏链的若干定理

发布时间：2018-06-22 00:16

本文选题：非齐次马氏链 + 几乎处处收敛　；参考：《江苏大学》2016年硕士论文

【摘要】：马氏信源作为信息论中的一种特殊信源,在信息论中一直具有重要地位,在解决人口增长问题、实际生活中的服务系统等领域起着重要的作用,信息论中的相对熵在群决策问题、图象算法问题、计算机系统中都起着重要的作用。然而在实际生活中,很多的信源不只是一重的,是二重甚至是多重的,这就有必要对二重马氏信源进行研究。故本文尝试将信息论中的结论推广到非齐次马氏链上,主要采用马氏链的极限定理和强大数定律这两个理论。本文首先介绍了本课题的研究背景,近期国内外作者的研究成果,并对本文的组织结构进行了简单规划,同时介绍了相关的概念、定理和性质;然后,本文利用鞅差序列的一个定理进而得到了非齐次马氏信源二元函数的极限定理(杨卫国),Csiszar I在信息论中证明了关于随机变量序列的一个极限定理,而该定理的推论就是相对熵。本文主要利用非齐次马氏信源二元函数的极限定理,将信息论中的定理推广到非齐次马氏链上。然而,我们发现在现实生活中,要想描述一类实际问题,往往要用到非齐次二重马氏链,而杨卫国利用鞅差序列的定理给出了非齐次马氏信源三元函数的一个极限定理,故而本文利用该定理将信息论中的定理继续进行推广。其次,Loève在概率论中给出了对独立随机变量序列的一个强极限定理进行了研究,而Shiryaev则利用Kolmogorov三级数定理对独立随机变量序列的一个新的强极限定理进行了证明,本文给出一个具体的例子,通过这个例子说明Shiryaev给出的是一个新的类型的定理。最后,对文章进行了归纳总结,并指出了在研究过程中可能存在的一些不足,同时也对以后的需要改进的方向进行简单介绍。
[Abstract]:As a special information source in information theory, Markov information source has always played an important role in solving the problem of population growth and service system in real life. Relative entropy in information theory plays an important role in group decision making problem, image algorithm problem and computer system. However, in real life, many information sources are not only one, but even two or more, so it is necessary to study the double Markov sources. Therefore, this paper attempts to extend the conclusions of information theory to non-homogeneous Markov chains, mainly using the limit theorem of Markov chains and the strong law of numbers. This paper first introduces the research background of this topic, the recent research results of domestic and foreign authors, and the organization structure of this paper, and introduces the related concepts, theorems and properties. In this paper, by using a theorem of martingale difference sequence, we obtain the limit theorem of non-homogeneous Markov source binary function (Yang Weiguo). In information theory, I proves a limit theorem on the sequence of random variables, and the corollary of this theorem is relative entropy. In this paper, by using the limit theorem of binary functions of inhomogeneous Markov sources, the theorems in information theory are extended to non-homogeneous Markov chains. However, we find that in real life, in order to describe a class of practical problems, non-homogeneous double Markov chains are often used, and Yang Weiguo gives a limit theorem of non-homogeneous Markov source ternary functions by using the theorem of martingale difference sequence. Therefore, this paper uses the theorem to continue to generalize the theorems in information theory. Secondly, in probability theory, Lo 猫 ve studies a strong limit theorem for the sequence of independent random variables, while Shiryaev proves a new strong limit theorem for the sequence of independent random variables by using Kolmogorov's third-order number theorem. In this paper, a concrete example is given to show that Shiryaev is a new type theorem. Finally, the article is summarized, and some shortcomings in the research process are pointed out. At the same time, the direction of improvement in the future is also briefly introduced.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O211.4

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