The Strong Law Of Large Numbers For Multivariate Functions Of Continuous-State Nonhomogeneous Markov Chains

Markov chain is one of the most important stochastic process in the study of probability theory,which is widely used in the fields of many social sciences.Scholars have conducted a lot of researches on the nonhomogeneous Markov chains,and there are fruitful results.This paper aims to explore the strong law of large numbers for multivariate functions of continuous-state nonhomogeneous Markov chains.This paper consists of five chapters.In the first chapter,the paper mainly introduces the research background and research status of Markov chains.And the main research content and the arrangement of subsequent chapters are introduced.In the second chapter,some definitions of discrete-state Markov chains are introduced.Then the definitions of continuous-state Markov chains are given.And the author introduces the convergence of random variable sequences,the definition and properties of conditional expectation,the definition and properties of martingale,which are the basic theoretical knowledge in the subsequent chapters.In the third chapter,the paper introduces some basic contents of ergodicity theorems for Markov chains.The author introduces the definitions of Dobrushin coefficient,geometric strongly ergodic,strongly ergodic and weakly ergodic for continuous-state homogeneous Markov chains.Then it gives a primary proof of equivalence of the ergodicities for continuous-state homogeneous Markov chains.In the fourth chapter,the author first studies the strong limit theorem of random variable sequence.On the basis of strong ergodicity of Markov chains,the strong law of large numbers for multivariate functions of continuous-state nonhomogeneous Markov chains are obtained.Finally,as the application of the strong law of large numbers,we get two corollaries.In the last chapter,the author summarizes the full text,points out the shortcomings and improvements in the research process.Meanwhile,the author indicates the future research content and exploration direction.