The Strong Law Of Large Numbers For The Delay Mean Of Frequencies Of Occurrence Of Countable States For Homogeneous Markov Chains

Probability theory is a discipline which studies the statistical law of random variables.It is applied to various fields such as information theory,insurance and risk assessment amd so on.Markov process is an important random process,which has a very profound theoretical basis and wide application space.Therefore,it is very meaningful to study the limit theory of Markov chains.As an important branch of mathematics,Markov chains still have research value and is an important subject.Literature[34]studies the generalized entropy ergodic theorem for nonhomogeneous Markov chains with finite state space,this paper main studies the generalized entropy ergodic theorem for homogeneous Markov chains with countable state space.The main contents are as follows:The first chapter of this paper is the introduction,mainly introduces the current research background and theoretical research progress of Markov chain,and gives the structure of this paper;The second chapter of this paper introduces the definition and the nature of several basic concepts,which is mainly to prepare some concepts and related knowledge involved in the proof of the theorem in this paper.Firstly,the definition and related properties of Markov chains are given and then introduces some basic concepts and properties of C-strong ergodicity,the smoothness of the conditional expectancy and condition expectation,and gives the relevant research results of the law of strong law of Markov chain.In this paper,the third chapter is the main content,we introduce the strong law of large numbers of the frequencies of state for the delayed sums in the countable state space.Because of the state space is countable,the operation for the countable sum and the limit can not be exchanged.So in this paper,we give a lemma firstly,then we use the smoothness of the condition expectation and the strong limit theorem of the binary function for delayed sums for proving the conclusion fo this paper,that is the limit of S_an^an+Φ（n）（i;ω）/(Φ（n）The fourth chapter of this paper is mainly to promote the conclusions of the third chapter to the strong law of large numbers of the frequencies of occurence of the ordered couples of states for the delayed sums.The lemma of the third chapter is applied to prove the corresponding conclusion,that is the limit of S_an^an+Φ（n）（i;j;ω）/(Φ（n）...