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(Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering,School of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410004,China;School of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410004,China)

机构地区：[1]Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering,School of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410004,China [2]School of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410004,China

出　　处：《Acta Mathematica Scientia》2022年第3期957-974,共18页数学物理学报（B辑英文版）

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)。

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.

关 键 词：Branching process with immigration random environment convergence rates central limit theorem convergence in law convergence in probability 

分 类 号：O211.6[理学—概率论与数理统计]