Asymptotic Propertyes Of Branching Processes With Immigration In Random Environments

The branching processes with immigration in random environment s is one of the important research directions of probability theory.With the development of the relevant theories of the branching processes,the derivative p roblems of such problems have also received extensive attention from scholars.In recent years,with the development of branching processes,many scholars have generalized this model,such as controlled branching processes,weighted branching processes and branchi ng random walks.Its application is very broad and has played an important role in the fields of physics,biology and genetics.Many phenomena in the natural environment can be studied by branching process,such as cell asexual reproduction,animal and pla nt reproduction,etc.,Such models provide a systematic theoretical method for the evolution of species,and play an important theoretical guiding role in many fields such as biomedicine,ecology,p hysics,etc.Therefore,it has very practical research b ac kground and application value.The action of adding the Poisson process as immigration variable to the branching process model makes such models gradually become an important research direction in this field.Based on previous studies,this paper mainly st udies the asymptotic properties and limit behaviors of the branching processes with immigration in random environment s,including moment properties,large deviation s,and central limit theorem s.In the study of such models,the interesting issues are the a symptotic properties of the population size and its limit theor y.For this purpose,the asymptotic propert ies of the moment of sub-martingale are need ed.Aiming at the branching processes with immigration in random environment s,firstly,this paper studies the uniform boundedness of moment s of sub-martingale: for positive moment s,we put forward the conditions for its uniform boundedness.For the harmonic momen ts,we give the order range of its uniform boundedness.Secondly,using the relevant conclusions o f the martingale and the s ub-martingale,we fo cus on the asymptotic properties for moment s of the population size,and stud y the precise converge nce rates of positive and harmonic moment s,resp ectively.Finally,based on the above theories of martingale and population size,the large deviation s and the convergence rate of central limit theorem s are studied.In terms of large deviation s,the prin ciple of large deviation of this model is established.Then the speed fun ctions of the upper and lower deviation are given respectively.In terms of the central limit theorem s,the Berry-Esseen bound theory is given.The branching processes in random enviro nments has m ore practical research significance than the classic Galton-Watson processes.Because of a biological point of view,environmental factors play a vital role in the reproduction of a population.However,the addition of environmental variables has also ma de the study of such models more complicated.The research in this paper enriches and improves the co ntent of the branching processes in rando m environment s.Fu rthemore,our research lays a theoretical foundation for the application of cell conta mination mo del and other biological models.