Strong Approximation Theorems For A Class Of Random Sequences

The original model of Markov process is Markov chain,which is proposed by Russian mathematician A.A.Markov in 1907.The major difference of Markov chain compared with other random processes is the non-aftereffect property of Markov chain,which means under current conditions,the probability distribution of future state is not directly related to the past state.This character can be seen as a generalization of independent random sequences in the probability theory.Markov process is an important branch of stochastic process.It plays not only an important role in the research of probability theory,but also has wide applications in the fields of modern physics,queuing theory,communication,social science,control,computer and finance.In 1983,Alam and Joag-Dev introduced the concept of NA random variable sequence.Due to the application in the limit theory and statistics,which were attracted wide attention from worldwide scholars,and some important progress have been made.In this paper,the concepts of sliding likelihood ratio and sliding relative entropy of random variable sequence of M values are introduced.Based on the two concepts and B-C lemma,we present a strong limit theorem for the moving average of M value random sequence and its corollary inference.The "random field" was born in the last thirty years,which is an interdisciplinary subject of probability theory and statistical physics.On the one hand,it provides a strict mathematical tool for statistical physics.On the other hand,it has greatly opened up the research field of probability theory.We usually divide the random field into a random field on a grid and a random field on a tree map.The important content of the airport is the Markov random field on the grid and the tree graph.In this paper,our research focus is on strong approximation theorem for a class of random sequences.We introduce the concept of moving average,likelihood ratio and martingale,and the pure analysis method to extend the strong approximation theorem of random sequences,and get the related results.This paper is divided into six chapters.The first chapter is the introduction part,which introduces the current research situation globally,research background,research methods and the main problems to be solved.The second chapter covers the basic theories and concepts,which list the relevant concepts used in the paper and theoretical knowledge.The third chapter presents a random sequence NA strong limittheorem.The fourth chapter introduces the concepts of sliding likelihood ratio and moving relative entropy.By constructing a parametric generalized likelihood ratio function,a strong limit theorem of random sequence of sliding average and main conclusions are obtained.The fifth chapter of the paper further introduces the asymptotic log likelihood ratio and the construction of martingale.A strong the deviation on spherical symmetry tree index of Markov chain(also known as the small deviation theorem)is presented,and partial results obtained are used to generalize a known.The sixth chapter covers conclusion and future prospective.