Convergence Rates Of The Branching Model

In recent years,the branching model has obtained a series of results in its extinction problem,asymptotic properties,Berry-Esseen bound and central limit theorem.In this paper,we studied the convergence rates,LP convergence and central limit theorem for branching process with immigration in a random environment,and the rate of convergence a.s.or in law for a multi-type branching process.This paper is divided into five chapters:In the first chapter,the research status of the convergence rates of branching model are summarized.The model and relevant definitions of branching process with immigration in a random or varying environment and multi-type branching process are introduced.The main results in this paper are given.In the second chapter,we are interested in the a.s.convergence rate of the submartingale Wn=Zn/Пn to its limit W,where(Пn)is an usually used norming sequence.The result about convergence almost sure(a.s.)are as following.Under a moment condition of order p∈(1,2)and limn→∞logmn/n=0 a.s.,where mn=EYn,W-Wn=o(e-na)a.s.for some a>0 that we find explicity;assuming EW1logW1α+1<∞ for some a>0,we have W-Wn=o(n-α)a.s.;similar results hold in a varying environment,but the condition limn→∞longmn/n=0 a.s.will be replaced by∑n=0∞anmn/Πnmn<∞ where(an)is a positive sequence of real numbers.In the third chapter,we studied the following central limit theorems and results about the rates of convergence in probability or in law for a branching process in a random environment.(ⅰ)W-Wn with suitable normalization converges to the normal law N(0,1);similar results also hold for Wn+k-Wn for each fixed k∈N*.(ⅱ)For a branching process with immigration in a finite state random environment,if W1 has a finite exponential moment,then so does W,and the decay rate of P(|W-Wn|>ε)is supergeometric.(ⅲ)There are norming constants an(ξ)(that we calculate explicitly)such that an(ξ)(W-Wn)converges in law to a mixture of the Gaussian law.In the fourth chapter,we show a necessary condition and a sufficient condition for the quenched Lp(p>1)convergence of(Wn).We then show that the convergence rate is exponential,and we find the maximal value of p>1 such that pn(W-Wn)→0 in Lp,in annealed sense.Similar results are also shown for a branching process with immigration in a varying environment.In the fifth chapter,we study the convergence rates of W(k)-Wn(k)a.s.convergence under the condition of different moment,and consider the convergence rate of W(k)Wn(k)converges to a mixture of the Gaussian law by proper normalization under the condition of second moment for a multi-type branching process.