The Study Of The Problems Ralated To Branching Processes With Exchangeable Environments

Branching process is a kind of stochastic processes, in the real life, there aremany phenomena can use a branching process mathematical model to describe, so ithas very important significance to research the branching processes. But with thepassage of time and the development of society, this classic branching processtheory shows some deficiency. Random environment branching processes have insome sense made up of classical branching process in a practical limitation onapplication. The theory of the branching processes in random environments ( BPRE )is the simulation of the method of population evolution and degeneration, theestablishment of BPRE enrich modern branching theory, with the rapid developmentof science and technology, it will play a more important role in other fields.The theory of branching processes in random environments is an active andfruitful branch in the probability theory research. In this paper, under the guidanceof my tutor I mainly extend some results in the identify environment to theexchangeable environment, there are four chapters.Chapter one: Introduction．The research significance of the branching processesand branching processes in random environments and the domestic and foreignresearch to this situation have been introduced, and the branching process model inrandom environments has been described with the precise mathematical language,the main results of this thesis have been summarized.Chapter two: Ratio Theorem. It gives some results of the problems related to theRatio Theorem in the exchangeable environments．Chapter three: Probability Generating Function．It studies some properties ofthe probability generating function and its inverse function in random environments．Chapter four: The asymptotic properties of survival probability of CriticalBranching Processes in Random Environments．The asymptotic properties ofsurvival probability of Critical Branching Processes under the specific conditions inRandom Environments have been discusses．...