CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

作     者：Yingqiu LI Xulan HUANG Zhaohui PENG 李应求;黄绪兰;彭朝晖

作者机构：Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in EngineeringSchool of Mathematics and StatisticsChangsha University of Science and TechnologyChangsha 410004China School of Mathematics and StatisticsChangsha University of Science and TechnologyChangsha 410004China 

出 版 物：《Acta Mathematica Scientia》 (数学物理学报（B辑英文版）)

年 卷 期：2022年第42卷第3期

页      面：957-974页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

基　　金：supported by the National Natural Science Foundation of China(11571052,11731012) the Hunan Provincial Natural Science Foundation of China(2018JJ2417) the Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering(2018MMAEZD02) 

主　　题：Branching process with immigration random environment convergence rates central limit theorem convergence in law convergence in probability 

摘      要：We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.