Strong Limit Theorems For A Class Of Nonhomogeneous Markov Chains

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.