On General Asymptotic Equipartition Theorems For Non-homogeneous Markov Chains And Some Applications

As a fundamental and ancient branch of mathematics, Probability Theory early mainly study the possibility of problems in life. With the deepening of its theory and reforming of research methods, it has been widely used in the financial, medical, military, and even the social sciences. In the processing, some Borderline Science has been formed, like materialism, biostatistics, and actuarial science. Markov processes play extraordinary role in financial decision-making, numerical calculation, management science, etc. The limit theory of Markov chains is one of the basic field theories in Markov process, it is so many scholars that have had a strong interest on the theory of Markov chains and their applications. In recent decades, the scholars have acquired some better results in researching on non-homogeneous Markov chain limit theorem. In the 1980 s, Professor LIU Wen and YANG Weiguo proposed a new way to study the strong limit theorem, some others also used this method to study the strong law heterogeneous Markov chain, strong deviation theorem, entropy theorem and other issues.This paper can be divided into two parts, focusing on the theorem of heterogeneous Markov chain generalized asymptotic average and its application. The first part studies a few limit theorems of random variables, including strong deviation theorems in the form of inequality. The second part mainly studies the broad asymptotic equipartition property(AEP) of m-order non-homogeneous Markov information source and its application. This paper will be divided into five chapters. In the introduction, we firstly introduces the development and the background of the theory of probability limit theory,and briefly describe the basic ideas and methods of strong deviation theorems of random variables founded by Liu, and finally introduces this paper’s researching methods.The first chapter introduces the entropy theorem of moving average for independent identically distributed random sequence and the central limit theorem based on Borel-Cantelli lemma theory and probability limit theory. Asymptotic equipartition theorem, a classical theorem on independent and identically distributed in probability theory, and the central limit theorem are the deduction of this chapter.The second chapter studies the generalized entropy of discrete source and generalized harmonic average a.s. convergence of random conditional probability theorem. A way will be put forward to make Markov inequality, Borel-Cantelli lemma and stochastic conditional moment generating functions and other tools applied in the research of strong limit theorem.The third chapter studies the broad asymptotic equipartition property(AEP) for m-order nonhomogeneous Markov information source and its application and obtains a class of strong limit theorem on such information source m+1 metafunction. And finally studies the broad asymptotic equipartition property(AEP) of m-order non-homogeneous Markov information source and its application.The fourth chapter puts forward the concept of broad empirical distribution function and studies its convergence.The fifth chapter introduces the concept of moving relative entropy, and uses it to study the dependence limit properties of discrete random variable sequence and finally gets the strong deviation theorems in the form of inequality. And then promotes some existing results.