Limit Properties Of Markov Chain In Random Environment

Many mathematicians pay attention to the theory of Markov chain in random environment (MCRE) by its widespread applications recent years.Nawrotzki and Cog-burn gave a general definition of MCRE , made the connection with this theory and the well-developed theory of Hopf Markov chains and got many important results on the theory of Markov chain in random environment.Orey made a survey on this problem and present a series of open problem.Kifer investigate the limit properties such as central limit theorem, law of iterated logarithm, large deviation of general MCRE. On the based of these literatures,The main object of this thesis is to devote to investigate the limit properties of Markov chain in random environment (MCRE). It includes the following parts:The classification of states of MCRE and the relationships of Markov chain in random environment with double Markov chains; The invariant measure and ergodicity of skew product Markov chain; The ergodic limit of Markov chain in random environment; The strong law of large numbers and the invariant principle of Markov chain in random environment, and the growth rate of multi-type branching process in random environment. In particular, we discuss the limit properties of MCRE when the random Doeblin condition is satisfied. There are seven chapters in this thesis.In chapter 2, we will discuss the Markov chain in random environment and the skew product Markov chain , p - 0 Hopf markov chain, which are generated by MCRE.At first,we give the definition of random transition matrix and MCRE, then construct the Markov chain in random environment (MCRE), the skew product Markov chain and p - 6 from a family of random transition matrices and a probability measure in infinitely dimensional space. Then we introduce Hopf Markov chain and discuss the relationship of MCRE,Hopf Markov chain and skew product chain.In chapter 3, we will discuss the characteristic and relationship of states in Markov chain in random environment. First of all, using the theory of general states Markov chain, we define some character number of skew product Markov chain. By these number, we introduce some basic concepts on the states, such as weak recurrent state, strongly recurrent state, and strongly transient state,essential state, proper essential state and then give some criteria.we prove that a strongly recurrent state is a weak recurrent state and that a strongly recurrent state is non-essential states. We also define "lead" and "uniform lead" between the states of MCRE. we prove that if x is a proper essential state and x can lead to y ,then y is also a proper essential state andif x is an essential state (or strongly recurrent state) and x can lead to y ,then y is an essential state (or strongly recurrent state), our paper provides the necessary and sufficient conditions for proper essential state where joint space is indecomposable and proper essential. At last, a example is present to show that weak recurrent state and strongly recurrent state are not equivalent in MCRE, which is equivalent in classical theory of Markov chain. Hence, it is meaningful for us to define the states of MCRE in the paper.In chapter 4, with the help of the ergodic theory of Hopf Markov chain, we will discuss invariant measure and ergodicity of skew product Markov chain. First of all, we introduce the basic theory of invariant measure and ergodic theory of Hopf Markov chain. Then we show that invariant measure skew product Markov chain exists if and only if the positive recurrent set is nonempty, any invariant measure is the linear combination of ergodic measure of Skew product, finally, the ergodic theorems of skew product Markov chain are obtained.In chapter 5, we use the properties of skew product Markov chain to investigate the ergodic limit of MCRE .At first,we introduce weak ergodicity and a partial ordering on the states of skew product markov chain, we find this ordering is equivalent on the positive set.Then discuss the relation between the weak ergodicity and a special equivalent relationship.when every...