The Study On Convergence Of Bisexual Galton-Watson Branching Processes With Immigration In Random Environment

The concept of bisexual Galton-Watson branching processes were first introduced by Daley in his paper "the Extinction conditions for certain bisexual Galton-Watson branching processes" in1968. Since then, many experts and scholars at home and abroad began to study and generalize this model. And obtained some generalized models such as bisexual Galton-Watson branching processes in random environment and the model of bisexual Galton-Watson branching processes with immigration etc.In this paper, under the guidance of my tutor, I mainly studied two different models of bisexual Galton-Watson branching processes with immigration in random environment, and obtained some related limit theorems. This thesis consists of three chapters.Chapter one:Introduction. We firstly gave a brief development history and applied backgrounds of bisexual Galton-Watson branching processes, then introduced three different generalized models:bisexual Galton-Watson branching processes in random environment, bisexual Galton-Watson branching with immigration of couples in random environment and the bisexual Galton-Watson branching Processes with immigration of females and males in random environment. Lastly, we gave the main research idea and the main results of this thesis in the end of this chapter.Chapter two:Galton-Watson branching with immigration of couples in random environment. We focused on some basic properties of the model and further study the convergence of the normalized processes, obtain the corresponding convergence theorems with respect to three different convergence types:almost sure convergence L1-convergence, L2-convergence.Chapter three:bisexual Galton-Watson branching Processes with immigration of females and males in random environment. We discussed bisexual Galton-Watson branching Processes with immigration of females and males in random environment. In two particular cases that mating flunctions are L(x,y)=ax+by-c and L(x,y)=xmin(1,y), we got some basi(results of the processes.