Branching Models In Random Environments And Characteristic Numbers For Birth-Death Processes With Barriers

Branching processes and birth-death processes, which are very important both in theory and application, form two classical and very active fields in probability theory. This thesis consists of two parts:the first part deals with branching models in ran-dom environments, which are natural and important extension of classic Galton-Watson branching processes, including from Chapter 2 to Chapter 6; the second part includ-ing chapter 7 studies the unified characteristic numbers of the birth-death processes with barriers, which is the basic work for the probabilistic construction of birth-death processes and further study the post-explosion properties.Chapter 1, we first give a brief development history and applied background of branching processes and birth-death processes respectively, then introduce the main results and structure of the thesis.Chapter 2, for a supercritical branching process (Zn) in a stationary and ergodic environmentξ, we focus on the rate of convergence of the normalized population Wn= Zn/E|Zn|ξ] to its limit W∞:we show a central limit theorem for W∞-Wn and Wn+k-Wn for each fixed k≥1 with suitable normalization and give a Berry-Esseen bound estimate for the rate of convergence in the central limit theorem when the environment is independent and identically distributed, then we discuss the corresponding iterated logarithm law. These results generalize the corresponding results in classic Galton-Watson branching processes.Chapter 3, We consider a branching random walk with a random environment in time, for A (?) R, let Zn(A) be the number of particles of generation n located in A. We obtain two results on central limit theorems for the counting measure Zn(·) with appropriate normalization, which generalize the corresponding results in classic branching random walk.Chapter 4, we first introduce the Sevast'yanov branching processes Z(t) in random environmentsξ, which is the generalization of the age-dependent branching processes in random environments. Then we study the corresponding integral equation of the conditional probability generating function E[sZ(t)|ξ], using the classical renewal the-orems. And we give the asymptotic properties of the mean of EZ(t), which show that EZ(t) grows with exponential rate under suitable condition. The results generalize the corresponding results in age-dependent branching processes in random environments.Chapter 5, we first introduce the branching process with random index in varying environments, which is composed of a branching process in varying environments and a renewal process. Let Yt and Yt denote the number of particles alive at time t and that of particles ever alive until time t respectively, and Zn be the embedded branching process in varying environments. We consider the problems of the extinction of Yt by the classic results of Zn, and show the finiteness of EYt and EYt under suitable condition, then further focus on the growth rate of EYt and EYt when EZn is regular varying in the supercritical case.Chapter 6, we consider concept of the recurrence and that of the transience in a Markov chains in random environments, show that various concepts of the recurrence and that of the transience are equivalent under suitable conditions respectively. In ad-dition, we give criteria for transience.Chapter 7, we first introduce the unified characteristic numbers for birth and death processes with barriers, then solve the related equations and express the solutions by unified characteristic numbers. This chapter is not only the basic work for solving probability construction problem of birth and death processes with leaping reflection barrier and quasi-leaping reflection barrier, but also is necessary for the further study the properties of post-explosion.