关于一类非齐次马氏链的强极限定理
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摘要
对马氏链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。有关齐次马氏链的研究已经形成了较完整的理论体系。实际数据的统计分析表明在大多数实际问题中随机系统的演化往往是时变的,必须考虑用非齐次马氏链来刻画,因此研究非齐次马氏链的极限理论是非常必要的,然而,由于缺乏适当的数学工具,关于非齐次马氏链的极限理论的研究结果并不多。本博士论文主要研究一类特殊非齐次马氏链一可列渐近循环马氏链的极限理论,我们证明了可列渐近循环马氏链的遍历性,强大数定律和熵定理,以及任意随机变量序列关于有限循环马氏链和有限m阶渐近循环马氏链的一类小偏差定理。还研究了关于时间离散状态连续非齐次马氏链的强偏差定理。
本博士论文共分六章。
第一章的第一节中介绍了马氏链的直观背景以及马氏链极限问题的研究进展。因为本文所研究的渐近循环马氏链是非齐次马氏链的一种特殊类型,所以在第二节中给出了关于非齐次马氏链的一些已知结果,如遍历性、强大数定律和强偏差定理,在论文的后续章节证明渐近循环马氏链的极限性质时,还会用到这些结论。第三节介绍了近年来关于渐近循环马氏链的研究成果,但这些研究仅局限在有限状态,本文将把其中的一些结果推广到可列状态。第四节详细的讲述了本论文的研究方法和主要结果。
第二章首先研究了可列渐近循环马氏链泛函的强大数定律,作为推论,得到一个众所周知的非齐次马氏链的强大数定律,并且得到了可列渐近循环马氏链关于状态出现频率的强大数定律。最后,给出了可列渐近循环马氏链的Shannon-McMillan定理。
第三章的第一节讨论了关于d步转移矩阵的C-强遍历性,然后给出了在此条件下可列非齐次循环马氏链的C-强遍历性。在第二节中首先研究了在强遍历的条件下,可列循环马氏链的收敛速度,然后进一步讨论了当满足不同条件时可列渐近循环马氏链的C-强遍历性、一致C-强遍历性和一致C-强遍历的收敛速度。
第四章用样本散度率距离作为随机变量序列相对于马氏链的差异的一种度量,通过限制此偏差,确定了概率空间的某子集,研究了在这个子集上任意随机变量序列关于有限循环马氏链的一类小偏差定理。
第五章利用m阶非齐次马氏链极限定理的已知结果,建立了任意随机变量序列关于m阶有限渐近循环马氏链状态出现频率的小偏差定理。作为推论,得到了这种马氏链的强大数定理和渐近均分割性。
第六章研究的对象是时间离散状态连续的非齐次马氏链,首先引入渐近对数似然比作为随机变量序列与马氏链之间偏差的度量,通过构造非负鞅,得到了时间离散状态连续非齐次马氏链的强偏差定理。
The study of limit theories for markov chain have grant significance and widely application in theory and practice.It is one of the main research projects in random process and limit theory. The research related with the homogeneous Markov chain has formed a relatively complete theoretical system. The statistical analysis of the actual data show that the evolution of a stochasis system in most practical problems are often time-varying, we must consider using a nonhomogeneous Markov Chain to describe it, so sthdy the limit theory of nonhomogeneous Markov chain is very necessary, however, because of the lack of appropriate mathematical tools, the result of the limit theory study for nonhomogeneous Markov chain is not much. This doctoral mainly studies the limit theory for a special kind of nonhomogeneous Markov chain—asymptotic circular Markov chain. We discuss the ergodic, strong law and entropy theorem of the asymptotic circular Markov chain, and a class of small deviation theorems of any random variables sequence and asymptotic circular Markov chains. We also study the strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains.
This doctoral dissertation consists of six chapters.
In the first section of chapter1, we introduce the intuitive background of Markov chains and the research progress of the limit property of Markov chains. The asymptotic circular Markov chains studied in this paper is a special type of homogeneous Markov chain, so in the second section gives some known results about nonhomogeneous Markov chains, such as ergodicity, the strong law of large numbers and the strong deviation theorem which may be used in the subsequent chapters of the paper to prove the asymptotic properties of asymptotic circular markov chains, and these conclusions may be used. In the third section, we introduced the research achievements of asymptotic circular markov chains in the recent years, but these studies are confined to the finite state, this article will extended some to the countable state. The fourth quarter presents the research methods and main results of this paper in details.
In chapter2, we study the strong law of large numbers for the functions of countable asymptotic circular Markov chains first. As corollary, we generalize a well-known version of the strong law of large numbers for nonhomogeneous Markov chains, and the strong law of large numbers on the frequences of occurrence of states for countable asymptotic circular Markov chains. Finally, we establish the Shannon-McMillan-Breiman theorem for this Markov chains。
The first quarter of chapter3, we discussed about the C-strong ergodicity of d-step transition matrix, and then give the C-strong ergodicity of countable nonhomogeneous circular Markov chain in terms of the C-strong ergodicity of d-step transition matrix. In the second section first studied the convergence rate of the countable circular Markov chains under the condition of strong ergodic, and then further discusses the C-strong ergodicity, the uniformly C-strong ergodicity of countable asymptotic circular Markov and the rate of convergence in Cesaro sense for countable asymptotic circular Markov under different conditions.
In chapter4, we use the samples divergence rate as a measure of difference between the sequence of random variables and Markov chains, is introduced. By restricting the deviation, a subset of sample space is determined, and on this subset the small deviation theorems for arbitrary random variables sequence on finite circular Markov chains are obtained.
In chapter5, we apply the known results of the limit theorem to mth-order nonhomogeneous Markov chains, the small deviation theorem on the frequencies of occurrence of states for mth-order asymptotic circular Markov chains is established. As corrally, the strong law of large numbers and asymptotic equipartition property for this Markov chain is obtained.
In chapter6, we study the discrete-time and continuous-state nonhomogeneous Markov chains. First, we introduce the notion of asymptotic average log-likelihood ratio, as a measure of the difference between the sequence of random variables and Markov chains, and by constructing a nonnegative martingale, the strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains is established.
本博士论文共分六章。
第一章的第一节中介绍了马氏链的直观背景以及马氏链极限问题的研究进展。因为本文所研究的渐近循环马氏链是非齐次马氏链的一种特殊类型,所以在第二节中给出了关于非齐次马氏链的一些已知结果,如遍历性、强大数定律和强偏差定理,在论文的后续章节证明渐近循环马氏链的极限性质时,还会用到这些结论。第三节介绍了近年来关于渐近循环马氏链的研究成果,但这些研究仅局限在有限状态,本文将把其中的一些结果推广到可列状态。第四节详细的讲述了本论文的研究方法和主要结果。
第二章首先研究了可列渐近循环马氏链泛函的强大数定律,作为推论,得到一个众所周知的非齐次马氏链的强大数定律,并且得到了可列渐近循环马氏链关于状态出现频率的强大数定律。最后,给出了可列渐近循环马氏链的Shannon-McMillan定理。
第三章的第一节讨论了关于d步转移矩阵的C-强遍历性,然后给出了在此条件下可列非齐次循环马氏链的C-强遍历性。在第二节中首先研究了在强遍历的条件下,可列循环马氏链的收敛速度,然后进一步讨论了当满足不同条件时可列渐近循环马氏链的C-强遍历性、一致C-强遍历性和一致C-强遍历的收敛速度。
第四章用样本散度率距离作为随机变量序列相对于马氏链的差异的一种度量,通过限制此偏差,确定了概率空间的某子集,研究了在这个子集上任意随机变量序列关于有限循环马氏链的一类小偏差定理。
第五章利用m阶非齐次马氏链极限定理的已知结果,建立了任意随机变量序列关于m阶有限渐近循环马氏链状态出现频率的小偏差定理。作为推论,得到了这种马氏链的强大数定理和渐近均分割性。
第六章研究的对象是时间离散状态连续的非齐次马氏链,首先引入渐近对数似然比作为随机变量序列与马氏链之间偏差的度量,通过构造非负鞅,得到了时间离散状态连续非齐次马氏链的强偏差定理。
The study of limit theories for markov chain have grant significance and widely application in theory and practice.It is one of the main research projects in random process and limit theory. The research related with the homogeneous Markov chain has formed a relatively complete theoretical system. The statistical analysis of the actual data show that the evolution of a stochasis system in most practical problems are often time-varying, we must consider using a nonhomogeneous Markov Chain to describe it, so sthdy the limit theory of nonhomogeneous Markov chain is very necessary, however, because of the lack of appropriate mathematical tools, the result of the limit theory study for nonhomogeneous Markov chain is not much. This doctoral mainly studies the limit theory for a special kind of nonhomogeneous Markov chain—asymptotic circular Markov chain. We discuss the ergodic, strong law and entropy theorem of the asymptotic circular Markov chain, and a class of small deviation theorems of any random variables sequence and asymptotic circular Markov chains. We also study the strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains.
This doctoral dissertation consists of six chapters.
In the first section of chapter1, we introduce the intuitive background of Markov chains and the research progress of the limit property of Markov chains. The asymptotic circular Markov chains studied in this paper is a special type of homogeneous Markov chain, so in the second section gives some known results about nonhomogeneous Markov chains, such as ergodicity, the strong law of large numbers and the strong deviation theorem which may be used in the subsequent chapters of the paper to prove the asymptotic properties of asymptotic circular markov chains, and these conclusions may be used. In the third section, we introduced the research achievements of asymptotic circular markov chains in the recent years, but these studies are confined to the finite state, this article will extended some to the countable state. The fourth quarter presents the research methods and main results of this paper in details.
In chapter2, we study the strong law of large numbers for the functions of countable asymptotic circular Markov chains first. As corollary, we generalize a well-known version of the strong law of large numbers for nonhomogeneous Markov chains, and the strong law of large numbers on the frequences of occurrence of states for countable asymptotic circular Markov chains. Finally, we establish the Shannon-McMillan-Breiman theorem for this Markov chains。
The first quarter of chapter3, we discussed about the C-strong ergodicity of d-step transition matrix, and then give the C-strong ergodicity of countable nonhomogeneous circular Markov chain in terms of the C-strong ergodicity of d-step transition matrix. In the second section first studied the convergence rate of the countable circular Markov chains under the condition of strong ergodic, and then further discusses the C-strong ergodicity, the uniformly C-strong ergodicity of countable asymptotic circular Markov and the rate of convergence in Cesaro sense for countable asymptotic circular Markov under different conditions.
In chapter4, we use the samples divergence rate as a measure of difference between the sequence of random variables and Markov chains, is introduced. By restricting the deviation, a subset of sample space is determined, and on this subset the small deviation theorems for arbitrary random variables sequence on finite circular Markov chains are obtained.
In chapter5, we apply the known results of the limit theorem to mth-order nonhomogeneous Markov chains, the small deviation theorem on the frequencies of occurrence of states for mth-order asymptotic circular Markov chains is established. As corrally, the strong law of large numbers and asymptotic equipartition property for this Markov chain is obtained.
In chapter6, we study the discrete-time and continuous-state nonhomogeneous Markov chains. First, we introduce the notion of asymptotic average log-likelihood ratio, as a measure of the difference between the sequence of random variables and Markov chains, and by constructing a nonnegative martingale, the strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains is established.
引文
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