李应求^1，*， 孙卫^1，， 刘伟^2，， 李德如^1，

（  1、长沙理工大学数学与计算科学学院,长沙,410014；       2、湖南大学工商管理学院,长沙,410082；  ）

摘要： 首先在过程本身的基本假设下,得出了随机环境中迁入分枝过程第n代粒子总数的条件期望和条件方差.其次,证明了随机环境中迁入分枝过程对应的规范化过程是一个非负下鞅.最后,在单个粒子产生后代数与迁入粒子数的方差和均值有限的条件下,推导出了随机环境中迁入分枝过程对应的规范化过程的几乎处处收敛性和L-2收敛性,这对于刻画过程本身的发散速度,具有重要的意义.
关键词： 迁入分枝过程; 随机环境; 下鞅; L-2收敛

Li Yingqiu^1,*， Sun Wei^1,， Liu Wei^2,， Li Deru^1,

（  1、College of Mathematics and Computing Scinece, Changsha University of Science and Technology,Changsha,410014；       2、School of Business, Hunan University,Changsha,410082；  ）

Abstract： In this paper, firstly, under the basic hypotheses of the process ,we deduced conditional expectation and variance of the total number of the n-th generation of particles in the branching process with immigration in random environment..Secondly, the normalization process corresponding the branching process with immigration in random environment is proved to be a nonnegative submartingale.Finally,under the condition of finite mean and variance, we get the limit properties of almost everywhere convergence and L-2convergence of process of normalization for branching process with immigration in random environment. It is very important for describing the rate of divergence for branching process with immigration in random environment..

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