广义生灭过程及序列的收敛性质
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摘要
在马尔可夫过程中，生灭过程无疑是其中极为重要的一类，但由于在许多现实模型中，具有状态空间E={0，1，2，…}的过程从任一状态i出发，下一步不但能到达相邻的状态i+1或i-1，且能回到初始状态0，从0出发可以达到任意状态，这时我们需要将一般生灭(拟)Q-矩阵，推广到更一般的具有更广泛的应用背景及理论研究价值的情况。本学位论文的第一部分，即第二、第三章致力于研究广义生灭(拟)Q-矩阵，其目的是获得易于检验的存在性、唯一性准则，以及获得诸如常返性、遍历性等重要概率性质。第四章研究马尔可夫Q过程的μ不变测度。
论文的第二部分研究序列的若干收敛性质，第五、第六、第七章致力于相依变量收敛性质的研究，获得了许多相依变量的与独立情形一样或接近的收敛性质。
第一章是绪论及介绍一些基本概念，主要是标准马尔可夫过程及各种混合序列、各种收敛性的一些基本定义、性质和一些基本关系。剩下的内容是本人在读博期间研究取得的主要结果，它们分为两部分：第一部分是广义生灭(拟)Q-矩阵和马尔可夫Q过程的μ不变测度，包括第二章、第三章和第四章；第二部分是序列的各种收敛性质，包括第五章、第六章和第七章。
第二章讨论具有突变率的广义生灭全稳定Q-矩阵，圆满地获得了Q-矩阵零流出、零流入以及Q过程的唯一性、常返性、遍历性、指数遍历性、强遍历性、随机单调性、Feller性、配称性等充分必要条件。
第三章讨论具有突变率的广义生灭单瞬时拟Q-矩阵，给出广义生灭拟Q-矩阵成为Q-矩阵的充分必要条件，且所有条件都是直接加在Q矩阵本身，因而易于验证、应用，并构造出全部Q过程和全部诚实Q过程，证明了所有诚实Q过程都是常返的。讨论并解决了过程的遍历性和可配称性，求出过程的遍历测度，最后证明了Kendall猜想对广义单瞬时生灭过程也成立。
第四章致力于p不变测度的研究，设m是Q的有限11不变测度，
在Q是全稳定、单瞬时不可和准保守拟Q－矩阵，以及含有吸收态
的情形，分别证明了存在Q过程P．t） 使m是Po）的人不变测度，
并且都具体构造出Q过程．
第五章得到了独立阵列和（含加权和）的最大值完全收敛的等
价条件，从而丰富和强化了前人的一系列结果．获得了负相关样
本线性模型中回归参数M估计是强相合的较弱的充分条件．
第六章研究两两NQD列的收敛性质，首先给出两两NQD列的
Kolmogorov型不等式，进而讨论它的若干收敛性质，获得了与独
立情形一样的h。和 Kat z完全收敛定理 2 几乎达到独立情形著
名的Mare ink柏呐cz 强大数定律，三级数定理，推广了著名的
Jam。son定理．
第七章研究两类较广泛的户混合序列及歹混合序列，给出歹
混合序列基本不等式，讨论并获得了户混合、歹混合序列的部分和
及加权和的收敛性质，推广了 Stout和 Thru。等定理．
birth-death processes is a greatly important kind of Markov processes. However, in most real applications, the processes with state space E = {0, 1,2,---} can make direct transitions from a state i not only to one of its nearest neighbor states i(if />i) and i + l, but also to state 0, from 0 to any state /. Therefore, we need to consider a new model(extended birth-death processes) with the special property in which catastrophes is imposed to ordinary birth-death processes. It is of considerably significance to study the model. The first part of dissertation, i.e., chapter 1 and chapter 2 is devoted to the studies on the model. The aims of the paper are to get some easy-checking criteria for existence and uniqueness and obtain important probability properties such as recurrence, ergodicity and equilibrium measure for the model.
Chapter 1 summarizes some basic concepts, mainly some basic definitions, properties and some basic relations in continuous-time Markov chains and all kind of mixing sequences. The rest of the paper is divided two parts: the first part, which is composed of chapter 2, chapter 3 and chapter 4, is devoted to studying problems on the extended birth-death processes and /^-invariant distribution. The second part is contributed to studying problems on convergence properties, which is composed of chapter 5, chapter 6 and chapter 7.
Chapter 2 is devoted to studying an extended birth-death stable Q -matrix with catastrophes. The necessary and sufficient conditions of uniqueness,recurrent, ergodicity, exponential ergodicity, strong ergodicity, stochastically monotone, Feller and symmetric properties
-in-
for the processes are presented.
Chapter 3 is devoted to studying an extended birth-death Q-matrix with catastrophes and instantaneous state. We are able to give easy-checking existence criteria for such processes. All the Q -processes and the honest Q -processes are explicitly constructed. Rcurrent and egodicity properties for the honest Q-processes are investigated. Surprisingly, it can be proved that all the honest Q-processes are recurrent without necessarily imposing any extra conditions. Ergodicity and symmetry of such processes are also investigated and solved. Equilibrium distributions are then established. Kendall's conjecture for the processes is proved to hold.
Chapter 4 is dedicated to the studies on //-invariant measure. Let m be a finite // -invariant measure of Q -matrix, as Q is totally stable, consists of a single absorbing state or single-instantaneous state, we prove that exists g-processes P(t], in which m be a ^ -invariant measure of P(t), and construct the Q -processes P(t).
Chapter 5 is focused on the studies on the equivalent conditions for maximum value convergence of sums of independent random matrix sequences, and the sufficiency condition of the strong consistency of M estimator of regression parametric in linear model for negatively associate samples, thus enriching and strengthening the results of a series of papers.
Chapter 6 is contributed to studying the convergence properties of pariwise NQD random sequences. We extend the Kolomogrov-type inequality, Baum and Katz complete convergence, the three series theorem, Marcinkiewicz strong law of large number and Jamison theorem.
Chapter 7 is dedicated to the study on the convergence properties of p mixing and p-mixing random sequences, discusses and obtains many convergence properties for the mixing random sequences, and extend the Stout and Thrumt heorem.
论文的第二部分研究序列的若干收敛性质，第五、第六、第七章致力于相依变量收敛性质的研究，获得了许多相依变量的与独立情形一样或接近的收敛性质。
第一章是绪论及介绍一些基本概念，主要是标准马尔可夫过程及各种混合序列、各种收敛性的一些基本定义、性质和一些基本关系。剩下的内容是本人在读博期间研究取得的主要结果，它们分为两部分：第一部分是广义生灭(拟)Q-矩阵和马尔可夫Q过程的μ不变测度，包括第二章、第三章和第四章；第二部分是序列的各种收敛性质，包括第五章、第六章和第七章。
第二章讨论具有突变率的广义生灭全稳定Q-矩阵，圆满地获得了Q-矩阵零流出、零流入以及Q过程的唯一性、常返性、遍历性、指数遍历性、强遍历性、随机单调性、Feller性、配称性等充分必要条件。
第三章讨论具有突变率的广义生灭单瞬时拟Q-矩阵，给出广义生灭拟Q-矩阵成为Q-矩阵的充分必要条件，且所有条件都是直接加在Q矩阵本身，因而易于验证、应用，并构造出全部Q过程和全部诚实Q过程，证明了所有诚实Q过程都是常返的。讨论并解决了过程的遍历性和可配称性，求出过程的遍历测度，最后证明了Kendall猜想对广义单瞬时生灭过程也成立。
第四章致力于p不变测度的研究，设m是Q的有限11不变测度，
在Q是全稳定、单瞬时不可和准保守拟Q－矩阵，以及含有吸收态
的情形，分别证明了存在Q过程P．t） 使m是Po）的人不变测度，
并且都具体构造出Q过程．
第五章得到了独立阵列和（含加权和）的最大值完全收敛的等
价条件，从而丰富和强化了前人的一系列结果．获得了负相关样
本线性模型中回归参数M估计是强相合的较弱的充分条件．
第六章研究两两NQD列的收敛性质，首先给出两两NQD列的
Kolmogorov型不等式，进而讨论它的若干收敛性质，获得了与独
立情形一样的h。和 Kat z完全收敛定理 2 几乎达到独立情形著
名的Mare ink柏呐cz 强大数定律，三级数定理，推广了著名的
Jam。son定理．
第七章研究两类较广泛的户混合序列及歹混合序列，给出歹
混合序列基本不等式，讨论并获得了户混合、歹混合序列的部分和
及加权和的收敛性质，推广了 Stout和 Thru。等定理．
birth-death processes is a greatly important kind of Markov processes. However, in most real applications, the processes with state space E = {0, 1,2,---} can make direct transitions from a state i not only to one of its nearest neighbor states i(if />i) and i + l, but also to state 0, from 0 to any state /. Therefore, we need to consider a new model(extended birth-death processes) with the special property in which catastrophes is imposed to ordinary birth-death processes. It is of considerably significance to study the model. The first part of dissertation, i.e., chapter 1 and chapter 2 is devoted to the studies on the model. The aims of the paper are to get some easy-checking criteria for existence and uniqueness and obtain important probability properties such as recurrence, ergodicity and equilibrium measure for the model.
Chapter 1 summarizes some basic concepts, mainly some basic definitions, properties and some basic relations in continuous-time Markov chains and all kind of mixing sequences. The rest of the paper is divided two parts: the first part, which is composed of chapter 2, chapter 3 and chapter 4, is devoted to studying problems on the extended birth-death processes and /^-invariant distribution. The second part is contributed to studying problems on convergence properties, which is composed of chapter 5, chapter 6 and chapter 7.
Chapter 2 is devoted to studying an extended birth-death stable Q -matrix with catastrophes. The necessary and sufficient conditions of uniqueness,recurrent, ergodicity, exponential ergodicity, strong ergodicity, stochastically monotone, Feller and symmetric properties
-in-
for the processes are presented.
Chapter 3 is devoted to studying an extended birth-death Q-matrix with catastrophes and instantaneous state. We are able to give easy-checking existence criteria for such processes. All the Q -processes and the honest Q -processes are explicitly constructed. Rcurrent and egodicity properties for the honest Q-processes are investigated. Surprisingly, it can be proved that all the honest Q-processes are recurrent without necessarily imposing any extra conditions. Ergodicity and symmetry of such processes are also investigated and solved. Equilibrium distributions are then established. Kendall's conjecture for the processes is proved to hold.
Chapter 4 is dedicated to the studies on //-invariant measure. Let m be a finite // -invariant measure of Q -matrix, as Q is totally stable, consists of a single absorbing state or single-instantaneous state, we prove that exists g-processes P(t], in which m be a ^ -invariant measure of P(t), and construct the Q -processes P(t).
Chapter 5 is focused on the studies on the equivalent conditions for maximum value convergence of sums of independent random matrix sequences, and the sufficiency condition of the strong consistency of M estimator of regression parametric in linear model for negatively associate samples, thus enriching and strengthening the results of a series of papers.
Chapter 6 is contributed to studying the convergence properties of pariwise NQD random sequences. We extend the Kolomogrov-type inequality, Baum and Katz complete convergence, the three series theorem, Marcinkiewicz strong law of large number and Jamison theorem.
Chapter 7 is dedicated to the study on the convergence properties of p mixing and p-mixing random sequences, discusses and obtains many convergence properties for the mixing random sequences, and extend the Stout and Thrumt heorem.
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