The Study Of Conditional Limiting Theorems And The Problems Related To Markov Chains

In this thesis, we study the conditional limiting theorems of Markov chains and the related problems. The study of quasi-stationarity and quasi-ergodicity for absorbing Markov chains have grant significance and widely application in theory and practice. It is one of the most active research projects of limit theory and random process. The quasi-stationary distribution and quasi-ergodic distribution bridge the gap between the known stationary behaviour and the unknown time dependent behaviour of a process. The Markov chain in this thesis generally does not satisfy usual stationarity condition, so we concentrate on their conditional stationarity, decay parameter and related problems. This doctoral dissertation consists of five chapters.In Chapter 1, we introduce the intuitive background of Markov chains and the research progress of quasi-stationary distribution and quasi-ergodic distribution of Markov chains. We introduce the basic model of absorbing Markov chains and give the definitions of quasi-stationarity, quasi-ergodicity and their domain of attraction. We also present the research methods and main results of this paper in details.In Chapter 2, the emphases are on quasi-stationary distributions and quasi-ergodic distributions. Our main tools in this chapter is "θK-classification". We study the existence, uniqueness and domain of attraction of the two limiting distri-butions for absorbing Markov chains. A sufficient condition for the existence and the uniqueness of quasi-stationary distributions and quasi-ergodic distributions is given in our main results and under this condition, the unique quasi-stationary distribution and quasi-ergodic distribution attract all initial distributions. Taking birth and death process as an example, we demonstrate the above results. We also show that the quasi-ergodic distribution is the stationary distribution of the limiting process which is an ergodic process. Finally, we discuss the stochastically monotone Markov chains.In Chapter 3, we consider the properties of eigenfunction, transition functions and quasi-ergodic distributions on birth and death chains. The main tools are the Karlin-McGregor decomposition and the dual process. We consider the birth and death process with entrance boundary and exit boundary. We prove that there is precisely one quasi-ergodic distribution if and only if the birth and death process is θK-positive recurrence. We shall discuss some conditions which are sufficient to deduce that the transition function is uniformly bounded of state. Finally, we study some important stochastic ordering relations for quasi-stationary distributions.In Chapter 4, we consider the decay parameter and "θK-classification" for a continuous-time Markov chain. We study the equivalent statement of decay pa-rameter and state some sufficient conditions for the positivity of decay parameter. Furthermore, we study the "θK-classification" of absorbing birth and death pro-cesses and linear birth and death processes with killing, giving sufficient conditions for such processes to be θK-positive recurrent.In Chapter 5, the dissertation finally summaries the results of this study, and points out the inadequacy and future perspectives on research work.