Limit Properties Of Multi-Type And Single-Type Branching Processes In Random Environments

In recent years,the research on the branching processes in the random environments has attracted the attention of many scholars.The multi-type branching process in random environment(MBPRE)and single-type branching process in the random enviro-nment(BPRE)are natural extensions of the branching process in definite environment.There have been many articles on these processes,due to numerous applications,for example in the description of the emergence and survival of mutant genes and in analyzing the fluctuation phenomenon of queuing changes,in studying the continuation of family surname,population growth,particle fission and so on.Based on the existing results on the subject,this thesis further studies limit theorems on multi-type and single-type branching process in random environment.The thesis is divided into four chapters.In chapter 1,we present the background of MBPRE and BPRE and the research status in China and abroad.Chapter 2 concerns the research on the multi-type branching process in random environment.It is divided into five parts according to its content:the extinction problem,the gradual progress of extinction time,the supper-critical limit problem,a law of large numbers and an expansion problem.Based on the condition (?),we construct a martingale Wn(m)and discuss its limit.In chapter 3,we study the convergence rate of the multi-type branching process in random environment.Assuming that the limit of the martingale Wn(m)is non-degenerate.we use the method of Huang and Liu[24]to discuss the convergence rate of the martingale.In chapter 4,we study the central limit theorem of the single-type branching process in random environment,and use the method of corollary 3.2 of[8]by Comets and Liu.Based on the assumption that the Lindeberg condition is established,a central limit theorem for the single-type branching process in random environment is obtained.