【摘要】：馬爾可夫鏈?zhǔn)歉怕收撗芯恐械囊活愔匾碾S機(jī)過(guò)程,在計(jì)算科學(xué)、隨機(jī)分形、經(jīng)濟(jì)學(xué)、醫(yī)學(xué)、工業(yè)學(xué)等社會(huì)科學(xué)中有著廣泛的應(yīng)用。近年來(lái),汪忠志和楊衛(wèi)國(guó)在給出非齊次馬氏鏈的廣義熵密度之后,得到了關(guān)于非齊次馬氏鏈的一類極限定理即廣義熵遍歷定理。本文將運(yùn)用鞅方法將廣義熵遍歷定理分別推廣到一階非齊次馬氏鏈的一類二元函數(shù)上和二階非齊次馬氏信源上。首先,本文簡(jiǎn)略介紹了馬爾可夫過(guò)程的一些研究背景和國(guó)內(nèi)外的主要研究成就以及本文的結(jié)構(gòu)安排。隨后,本文給出了馬氏鏈和鞅論中的一些基礎(chǔ)知識(shí)以及馬氏鏈的研究過(guò)程中一些比較重要的引理。然后,本文首先介紹了非齊次馬氏鏈廣義熵密度的定義,以及楊衛(wèi)國(guó)得到的非齊次馬氏鏈的廣義熵遍歷定理。接著本文運(yùn)用鞅方法把一階非齊次馬氏鏈的極限定理推廣到一類函數(shù)上。此外,在生活中,我們往往要用二階馬氏信源去描述實(shí)際問(wèn)題。楊衛(wèi)國(guó)和劉文已經(jīng)得到了關(guān)于二階非齊次馬氏信源的經(jīng)典熵遍歷定理。為此,本文在給出二階非齊次馬氏信源的廣義熵密度的前提下,把廣義熵遍歷定理推廣到二階非齊次馬氏信源上。并得到二階非齊次馬氏信源的一類極限定理即二階非齊次馬氏信源的廣義熵遍歷定理。最后,對(duì)全文進(jìn)行了總結(jié),闡述了本文中的一些不足,并表明了以后的研究?jī)?nèi)容與探索方向。
[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.
【學(xué)位授予單位】：江蘇大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：F224

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