An Extended Birth-Death Process And Convergence Property For Sequences

birth-death processes is a greatly important kind of Markov processes. However, in most real applications, the processes with state space E = {0, 1,2,---} can make direct transitions from a state i not only to one of its nearest neighbor states i(if />i) and i + l, but also to state 0, from 0 to any state /. Therefore, we need to consider a new model(extended birth-death processes) with the special property in which catastrophes is imposed to ordinary birth-death processes. It is of considerably significance to study the model. The first part of dissertation, i.e., chapter 1 and chapter 2 is devoted to the studies on the model. The aims of the paper are to get some easy-checking criteria for existence and uniqueness and obtain important probability properties such as recurrence, ergodicity and equilibrium measure for the model.Chapter 1 summarizes some basic concepts, mainly some basic definitions, properties and some basic relations in continuous-time Markov chains and all kind of mixing sequences. The rest of the paper is divided two parts: the first part, which is composed of chapter 2, chapter 3 and chapter 4, is devoted to studying problems on the extended birth-death processes and /^-invariant distribution. The second part is contributed to studying problems on convergence properties, which is composed of chapter 5, chapter 6 and chapter 7.Chapter 2 is devoted to studying an extended birth-death stable Q -matrix with catastrophes. The necessary and sufficient conditions of uniqueness,recurrent, ergodicity, exponential ergodicity, strong ergodicity, stochastically monotone, Feller and symmetric properties-in-for the processes are presented.Chapter 3 is devoted to studying an extended birth-death Q-matrix with catastrophes and instantaneous state. We are able to give easy-checking existence criteria for such processes. All the Q -processes and the honest Q -processes are explicitly constructed. Rcurrent and egodicity properties for the honest Q-processes are investigated. Surprisingly, it can be proved that all the honest Q-processes are recurrent without necessarily imposing any extra conditions. Ergodicity and symmetry of such processes are also investigated and solved. Equilibrium distributions are then established. Kendall's conjecture for the processes is proved to hold.Chapter 4 is dedicated to the studies on //-invariant measure. Let m be a finite // -invariant measure of Q -matrix, as Q is totally stable, consists of a single absorbing state or single-instantaneous state, we prove that exists g-processes P(t], in which m be a ^ -invariant measure of P(t), and construct the Q -processes P(t).Chapter 5 is focused on the studies on the equivalent conditions for maximum value convergence of sums of independent random matrix sequences, and the sufficiency condition of the strong consistency of M estimator of regression parametric in linear model for negatively associate samples, thus enriching and strengthening the results of a series of papers.Chapter 6 is contributed to studying the convergence properties of pariwise NQD random sequences. We extend the Kolomogrov-type inequality, Baum and Katz complete convergence, the three series theorem, Marcinkiewicz strong law of large number and Jamison theorem.Chapter 7 is dedicated to the study on the convergence properties of p mixing and p-mixing random sequences, discusses and obtains many convergence properties for the mixing random sequences, and extend the Stout and Thrumt heorem.