【作者】 王豹；

【导师】 杨卫国；

【作者基本信息】 江苏大学 ， 系统工程， 2014， 博士

【摘要】 树指标随机过程是近年来概率论的研究方向之一,在许多学科中都有很好的应用,以生物学为例,生物学家在研究杆状菌分裂时,总结出杆状菌分裂的规律,即一个杆状菌在分裂时,从中间断开,这样就分裂成两个新杆状菌,这两个新的杆状菌称为原来的那个杆状菌的后代.如果把每一个杆状菌的寿命看作是一个随机变量,那么所有杆状菌寿命的全体就是一个树指标随机过程.对树指标随机过程的极限理论的研究是随机过程和极限理论中重要的研究课题之一,具有重要的理论意义和应用价值.近年来杨卫国教授采用与传统方法不同的研究方法,对树指标Markov链的强大数律与熵遍历定理等方面进行了一系列研究,在国内外重要学术刊物上发表了一系列论文.本博士论文在杨卫国的研究基础上,进一步研究了在可列状态空间取值的树指标Markov链的强大数定律与Shannon-McMillan定理.Course等人介绍了隐Markov树模型的概念,但是并不严格,鉴于隐Markov树模型在各学术领域有着非常丰富的应用,有必要为其建立起严格的数学定义及坚实的数学基础.本论文将对隐Markov树模型进行一系列的研究工作,主要集中在以下几个方面：首先由Course的描述,对隐Markov树模型给出严格的数学定义,进而得到其等价命题；然后得到隐Markov树模型的相关性质,以此为基础证明其存在性；最后证明Cayley树情形下隐Markov树模型的强大数定律及Shannon-McMillan定理.本博士论文共分五章.第一章,介绍了与本博士论文相关的理论背景及意义,对树指标Markov链的相关概念及已经取得的理论成果进行了回顾,介绍了经典隐Markov过程,在离散情况下,介绍了不同学科中所给出的隐Markov链的定义,并证明其等价性.第二章,第一节研究了取值于可列状态空间的Cayley树指标Markov链状态与状态序偶发生频率的强大数定律,所得结果推广了取值于有限状态空间Cayley树指标Markov链的相应结果；第二节研究了取值于可列状态空间的一致有界树指标Markov链的强大数定律.所得结果推广了取值于可列状态空间Cayley树指标Markov链的相应结果,并推广了取值于有限状态空间的一致有界树指标Markov链的相应结果.第三章,采用与第二章相类似的研究方法,得到了取值于可列状态空间Cayley树指标Markov链二元函数的强大数定律,作为推论得到了其状态与状态序偶发生频率的强大数定律及Shannon-McMillan定理.第四章,对隐Markov树模型进行一系列的研究工作,主要集中在以下几个方面：给出隐Markov树模型的严格定义,并据此得到一些基本性质及等价定义,做为推论,我们将给出隐Markov树模型的存在性的证明.第五章,研究了Cayley树指标下隐Markov树模型三元函数的强大数定律,做为推论,得到了Cayley树指标Markov链状态发生频率的强大数定律,关于隐Markov树模型参数的一些强大数定律及其Shannon-McMillan定理.更多还原

【Abstract】 The stochastic process indexed by a tree is one of the research directions in probability theory, and it has been widely applied to many disciplines. In the field of biology, for example, biologists summarized the rod-shaped bacteria’s division rules:one bacterium is devided into two equal parts, if every rod-shaped bacterium’s lifetime is considered as a random variable, the whole of them can be considered as a tree-indexed stochastic process, so the research of tree-indexed stochastic process is an important topic in the filed of stochastic process, and its limit theorems have very important theoretical and practical significance.In recent years, Professor Yang Weiguo has studied the strong laws of large numbers and Shannon-McMillan theorems for the tee-indexed Markov chains, a great deal of papers have been published in the national journals and international journals. This doctoral dissertation is on the basis of Yang’s research methods, further study the strong laws of large numbers and Shannon-McMillan theorems of the tee-indexed Markov chains taking values on countable space.Hidden Markov tree models were introduced by Crouse et al., but their definition is not strict. In this doctoral dissertation, we will give the rigorous definition for hidden Markov tree model, and obtain some properties for hidden Markov tree model, as a corollary, we will point out the existence of hidden Markov tree models. Finally we will obtain the strong laws of large numbers for hidden Markov tree models indexed by a Cayley tree, as corollaries, we will obtain some strong laws of large numbers for the parameters and the Shannon-McMillan theorem for this model.There are five chapters in this doctoral dissertation.In chaper1, we first give an introduction for the basic notations and review the theoretical results have been obtained about the tree-indexed Markov chains, then we will introduce the different definitions of hidden Markov chain in the different disciplines and prove their equivalence.In chapter2, the first secction, we study the strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a Cayley tree, which extend and improve the corresponding results in the finite states case. The second section, using the similar approach, we study the strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a uniformly bounded tree.In chapter3, we establish a strong law of large numbers for the bivariate functions of countable Markov chains indexed by a Cayley tree. As corollaries we get the strong laws of large numbers for frequencies of occurrence of states and ordered couples of states for countable Markov chain indexed by a Cayley tree, and get the Shannon-McMillan theorem for In chapter4, we study the hidden Markov tree model, give the vigorous definition for this model, and obtain some properties for them, as a corollary, we will prove the existence of hidden Markov tree models.In chapter5, we study the strong laws of large numbers for hidden Markov tree models indexed by a Cayley tree, and as collaries, we obtain some strong laws of large numbers for the parameters and the Shannon-McMillan theorem for this model.更多还原