Harmonic Moments And Berry-Esseen Bounds For Branching Processes In Random Environments

The model of branching process with immigration in random environments and branching random walks in random environments are extracted from specific biological and ecological examples.Due to its wide application value,it has attracted the attention of many scholars of probability theory and obtained abundant research results.In this paper,on the basis of the previous study of these two branching models by probability theorists,we are further discussed the existence of harmonic moments,Berry-Esseen bound and Cramér moderate deviation of this two processes.This paper is divided into four chapters.In the first chapter,we summarize the research background of branching model in random environments and introduce the current status of Berry-Esseen bound,harmonic moments,moderate deviation and large deviation.And then we give the strict definition of two branch models——the branching process with immigration in random environment and branching random walk in random environments.Finally,we give the main results of this paper.In the second chapter,we study the suppercritical branching processes with immigration in the environments.Considering the absence or omission of historical data,we use Minkowski inequality and H?lder inequality to proving the boundedness of E(Wn0,n),just so we give the Cramér moderate deviation of log zn0+n/zn0.And then we use the existence of harmonic moments of Wn to establish the Berry-Esseen bound of logzn0+n/zn0.From a statistical point of view,this result can be used as Zn0 to represent the confidence interval of the critical parameter μ.Similarly,the critical parameters μ,n and Zn0 can be used to estimate the size of Zn0+n.In which Wn=zn/Πn,Wn0,n≡(Wn0+n)/Wn0,Πn is a normalized sequence.In the third chapter,we study the branching random walk in a time-dependent random environment.Firstly,by calculating the Laplace transform of the limiting random variable W(t),we proved the the existence of harmonic moments of Wn(t).Secondly,we obtain the existence of moment of logW(t)and the exponential convergence rate of log Wn(t)converges to logW(t).Finally,the Berry-Esseen bound of log Zn(t)is established.In which Zn(t)=f e-txZn(dx),Wn(t)≡Zn(t)/Pn(t),Pn is a normalized sequence.The fourth chapter summarizes the main content of this paper and discusses the future research direction.