Invariant Distribution And Convergence Rate

Invariant distribution is a greatly important property of standard transition function in continuous-time Markov chains and jump processes, it is of considerably significnce to study it. Stochastic stability is of crucial importance in all kinds of stochastic models. Such investigation involves finding criteria of all kinds of convergence rates. The dissertation is devoted to the studies on invariant distribution and convergence rate. The aims of the paper are to identify invariant distribution of Q -function and to find criteria of convergence rate which are easy to check.The dissertation in chapter 1 summarizes some basic concepts, mainly some basic definitions, properties and some basic relations in continuous-time Markov chains and jump processes. The left is divided two parts. The first part, which is composed of chapter 3, chapter 4, is devoted to studying problems on invariant distribution and - invariant distribution. The second part is contributed to studying problems on convergence rate, which is composed of chapter 5, chapter 6, chapter 7.In chapter 3, first we answer the open problem of Williams(1979). We solve the problem completely when Q-matrix is totally stable or uni-instantaneous, that is, we not only prove the existence of Q -function but also identify the Q -function. Second, for invariant distribution of jump processes, we also obtain some good results.Chapter 4 is dedicated to the study on - invariant distribution. First, as Q -matrix is totally stable or uni-instantaneous, we prove the existence of - invariant distribution of Q - function and identify the Q - function. Second, we generalize - invariant distribution to jump processes and obtain some good properties and results.Chapter 5 is contributed to studying convergence rate of standard transition function and relation of all kinds of convergence rates. By revealing the close link among strong ergodicity, stochastically monotone, and the Feller transition functions we are able to prove that a monotone ergodic transition function is strongly ergodic if and only if it isnot a Feller transition function. An easy checking criterion for a minimal monotone transition function to be strongly ergodic is then obtained. We further prove that a non-minimal ergodic monotone transition function is always strongly ergodic.Chapter 6 is devoted to studying convergence rate of the birth and death processes. For a conservative birth and death Q -matrix, we prove that the minimal Q -function is strongly ergodic if and only if R - and S < . Suppose a birth and death Q -matrix satisfies R < and S < . Then there exists a unique reversible honest Q -function which is strongly ergodic.Chapter 7 is devoted to studying all kinds of the extended time-continuous branching processes. Ergodicity, exponential ergodicity and strong ergodicity criteria are presented. In the meantime, we also obtain the martingale property and the conditions of the upper and lower estimate of large deviations.