Large Deviations Of Branching Models In Random Environments And Related Problems

In an independent and identically distributed(i.i.d.)random environment,based on the family tree model of the branching process with immigration in the random envi-ronment,in this paper,we mainly study the moments,the estimates of large deviation,Berry-Esseen's bound of this process.The main innovation is to use the family tree model to deal with incoming particles in each generation.In addition,the moments,harmonic moments of the controlled branching process in the random environment are also studied.This paper is divided into three chapters:The first chapter is the introduction,the research background and significance of the branching process with immigration in the random environment and the controlled branching process in the random environment are firstly summarized.Then the strict definitions of the two models are introduced.Finally,the main research contents and results are given.In chapter 2,the asymptotic properties of the moments of the branching process with immigration in the random environment are concerned,the large deviation esti-mates is proved.Moreover,in the supercritical case,the Berry-Esseen's bound of the logarithmic process log Zn is studied.In addition,in the subcritical case,some initial value problems about the first passage time TtZ are also discussed.In chapter 3,the asymptotic properties of the moments and the existence of har-monic moments for controlled branching process in random environment are studied.