Limit Theorem Of Multiplicative Controlled Branching Process In A Random Environment

In response to the realistic requirement of controlling the size of offspring for particle reproduction,Dion and Essebbar(1995)introduced a multiplicative controlled branching process(MCBP).At the same time,since particle reproduction is affected by various environmental factors,the multiplicative controlled branching process in a random environment(MCBPRE)has important theoretical and practical significance.This thesis mainly studies the limit theorem problems of MCBPRE,such as convergence rate,Lp convergence,harmonic moment and small value probability.There are five chapters in this thesis.The first chapter is introduction,which outlines the research background and research status,introduces the definition of MCBPRE and multiplicative controlled branching process in a varying environment(MCBPVE),and gives the main results of this thesis.The second chapter mainly studies the a.s.convergence rate of the normalization process {Wn}n≥0 to W.It is proved that under the condition of moment of order p∈(1,2),a certain a>0 can be found such that W-Wn=o(e-na)a.s.and suppose that for some α>0,EW1 log W1α+1<∞,then W-Wn=o(n-α)a.s.holds.Furthermore,in a varying environment,similar results can be obtained.The third chapter obtains the results of the in probability convergence rate and in distribution convergence rate.First,it is proved that under the condition of the second moment,there exists a normalization constant an(ξ),which makes an(ξ)(W-Wn)in law converge to a non-degenerate distribution.Secondly,it is proved that in the case where the environment ξtakes a finite state space,if the exponential conditional moment of Wi is limited,then the exponential conditional moment of W is also limited,and The decay rate of P(|W-Wn|>ε)is hypergeometric.The fourth chapter mainly studies the convergence of Lp and the convergence rate of Lp,and obtained the four equivalent conditions for the convergence of quenched Lp and sufficient conditions for the exponential convergence rate of Lp in a varying environments.The fifth chapter mainly studies the harmonic moment and the probability of small value.We get the harmonic moment when the initial particle is k and for any j≥k,when n→∞,the exact decay rate for the probability P([αnZn]|Z0=k).