The Multiplicative Controlled Branching Process In A Random Environment

In 1975,Dion and Essebbar[1]introduced the multiplicative controlled branching process,which is a special case of the controlled branching process.Considering that the change of the epidemic situation is affected by the environment,today,when the COVID-19 is increasingly controlled,the multiplicative controlled branching process in a random environment is more meaningful.Therefore,the definition and convergence,existence of weighted moments,central limit theorem and Berry-Esseen bounds of the multiplicative controlled branching process ?Zn?n?0 are introduced in this paper.There are four chapters in this paper:The first chapter illustrates the research background and current status,and describes the concepts of the multiplicative controlled branching process in a random environment and varying environment,and gives the primary consequences of this paper.In the second chapter,we study the convergence of the multiplicative controlled branching process {Zn}n?0 in a random environment.A sufficient condition for the normalized process {Wn} which converges almost surely to a finite random variable W is given by the supermartingale convergence theorem.We discuss the necessary and sufficient condition for L1-convergence and the necessary condition for L2-convergence about {Wn}.It is proved that the Lp(p?1)convergence of {Wn} by proving that {Wn} is the fundamental sequence of Lp-convergence,where,Wn=Zn/Pn,(?)#12In the third chapter,an inequality of Asussen and Hering[2]is generalized,and a sufficient condition for the existence of quenched weighted moment of W is given by using this generalized inequality.In the fourth chapter,we research the central limit theorem of log Zn and the Berry-Esseen bounds.We apply the exponential speed of convergence of log Wn to log W to the Berry-Esseen bounds of log Zn.