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几类马氏骨架过程的研究与Q过程的若干性质
The Study of Several Classes of Markov Skeleton Processes and Some Properties for Q-processes
【作者】 唐荣；
【导师】 刘再明；
【作者基本信息】 中南大学 ， 概率论与数理统计， 2005， 博士
【摘要】 马尔可夫过程是一类重要的随机过程,它的原始模型马尔可夫链,由俄国数学家A.A.马尔可夫于1907年提出。马尔可夫过程的“核心”是马尔可夫性,其直观描述是:在已知系统的目前状态的条件下,系统未来的演变不依赖于它以往的演变。可列马尔可夫过程是马尔可夫过程的一个非常活跃而且研究成果非常丰富的分支,如文献[1]～[4]等。其中,积分型随机泛函、数字特征以及Q矩阵问题的研究是其重要的研究内容。 马尔可夫骨架过程是在一列停时处具有马氏性的随机过程。它较马氏过程、半马氏过程更为广泛,为一类更广泛的实际问题提供了随机模型。马尔可夫骨架过程是由侯振挺教授等人于1997年首先提出,随后他和他的学生们在这一领域开展了卓有成效的工作,发表了一系列文章并出版了专著,如文献[6]等。 本文的研究包含两部分内容:一是研究几类马尔可夫骨架过程,其中包括:半马氏过程,半马氏生灭过程以及生灭型半马氏骨架过程;二是讨论了Q过程的一些重要性质,并构造了一类全稳定Q过程。全文共分七章,主要结果有: 1.研究了半马氏过程的一维分布,构造及积分型随机泛函。 2.给出了半马氏生灭过程的定义,引进了其数字特征,讨论了向上和向下积分随机泛函、遍历性及平稳分布。 3 提出了生灭型半马氏骨架过程的定义,求出了两骨架时τn-1（ω）与τn（ω）之间的嵌入过程X（n）（t,ω）的初始分布及寿命分布,得到了生灭型半马氏骨架过程的一维分布,构造了生灭型半马氏骨架过程,引进了生灭型半马氏骨架过程的数字特征并讨论了它们的概率意义,最后讨论了向上和向下的积分型随机泛函。 4.研究了马氏过程P（t）的分解及p∞j（s,t）的定义,Q过程的B条件成立的充分必要条件,引进了数字特征并讨论了其概率意义,研究了Q过程的积分型随机泛函,引进了极小过程的概念,得到了两个解析结构定理。 5.引进了向后首达时间和向前道达时间,讨论了他们的分布及性质,得到了向前禁止概率分解定理和向后禁止概率分解定理。 6.讨论了Q矩阵问题。我们得到了全稳定Q过程构造的等价条件。构造出了
【Abstract】 Markov processes is one class of important stochastic processes. Markov chain which is proposed by Russian mathematician A.A. Markov in 1907 is its primitive model.The core of Markov processes is Markov property which can be described as: known current state condition,the change of Markov processes in the future will not depend on previous change.Countable Markov processes is an active branch of Markov processes, and its research fruit is very abundant.The study of integral type random functional, numerical characteristic and Q-matrix is countable Markov processes important research contents.The Markov skeleton processes is stochastic processes which has Markov property at stopping time. Its area is more extensive than Markov processes and semi Markov processes and provide random models to more extensive applied problem. The Markov skeleton processes is proposed by professor Hou Zhengting etc in 1997.He and his students get fruitful works in this field.A series of articles and bibliographies have been published,for example bibliography [6],etc.The study of this thesis includes two aspects.One is to study several classes of Markov skeleton processes which include the semi Markov processes,the birth and death semi Markov processes,the birth and death type semi Markov skeleton processes, the other is to discuss important theories of the Q-processes and to construct one class of all-stable Q-processes.It consists of seven chapters.We drew the following conclusions.1. We study the one-dimensional distribution,integral type random functional and construction of the semi Markov processes.2. We introduce the concept of the birth and death semi Markov processes,and its numerals characteristics,and discuss up and down integral random functionals meanwhile.3. We introduce the definition of the birth and death type semi Markov skeleton processes,and get the initial distribution and the life distribution of Xn（t,w）which imbed between skeleton sequence time rni（u;） and rn（w）.We get the one-dimensional distribution of the birth and death type semi Markov skeleton processes. We make use of the one-dimensional distribution and initials distribution to construct the birth and death type semi Markov skeleton processes. We introduce the numerals characteristics of the birth and death type semi Markov skeleton processes ,and discuss their probability meanings.In finally,we study the up and down integral type random functionals.4. We study the decomposition of Markov processes and give the definition of Pooj（t）.The necessary and sufficient condition of the Q-processes’s B-condition is given.We introduce in Q-processes’s figure characteristics ,and discuss probability meanings of the figure characteristics .We study the integral random functional of the Q-processes.By introducing the concept of the minimal processes,we have obtained two analytic construction theorems.5. We introduce the backward first arrival time and the forward first arrival time,and discuss their probability distributions and properties.Finally,we get the decomposition theorems of the forward taboo probability and the backward taboo probability.6. Q-matrix problem is discussed.We obtain equivalent conditions of Q-processes structure,and construct all-stable Q-processes and all-stable honest Q-processes about the same flow into on A（i）.We get honest Q-processes’s criterion of the existence.Finally,we get the analytical expression of （Q, II） processes.