The Normal Approximation Of The Stochastic Index Branching Process

Branching process in a random environment(BPRE)is one of the important research fields in probability theory.It is widely used in biology,physics,engineering,economics,etc.Usually,subject to the influence of various factors in the space,the environment of the particles is also changing,so compared with the classical branching process,BPRE can describe the variation of the particles more accurately.Poisson randomly indexed branching process(PRIBP)studied in this paper essentially is a BPRE.In branching process,estimation of offspring mean m is one of the key points,one of the most important statistics is Lotka-Nagaev estimate.How to measure the error between the estimator and m is one of the main problems we interest in.The main tools for measuring error are normal deviation,large deviation and moderate deviation.This paper mainly focuses on the normal deviation.For the normal deviations of Lotka-agaev estimation of the classical branching process,when the deviations are constants,the limit distribution is non-normal and it was obtained by Nagaev in 1967,when the deviations are random variables,the limit distribution is normal and it was obtained by Dion in 1974,Berry-Esseen bound was obtained by Heyde in 1971.Later,Berry-Esseen estimation and LIL of BPRE are obtained by Gao and Hu in 2012.In this paper,we consider the asymptotic distribution and the Berry-Esseen bound of PRIBP.The structure of the article is as follows:In the first chapter,we briefly introduce the basic knowledge of the classical Galton-Watson branching process and the research progress of the normal deviation of Lotka-Nagaev estimator.Then the main results of this paper are given.In the second chapter,we study dacay rate of the second-order moments and asymptotic distribution of the Lotka-Nagaev estimator normalized by a non-random sequence.Decay rates of the second order moments play an important role in the proofs.In the third chapter,we study the asymptotic normality of the Lotka-Nagaev estimator and Berry-Esseen bound.The main tool we use in the proof is harmonic moments of PRIBP.Finally,in the last chapter,the method is used to study the martingale generated by PRIBP.Then,we give some contents that we will study recentlly.