带移民的上临界分支过程在一般条件下的小值概率

Small value probabilities for supercritical branching processes with immigration in general case

本文的主要目的是在后代分布均值有限但L log L阶距无限的条件下研究带移民的上临界分支过程(Z_n)的小值概率.当后代分布均值有限且移民分布的log L阶距有限时,存在常数序列{C_n,n≥0}使得C_n^(-1)Z_n收敛到一个非负有限且非退化的随机变量,记作W.本文基于前期关于分支过程小值概率的工作,在最一般的条件下得到了W的小值概率,即P(W≤ε)在ε→0^+时的收敛速率.

The main purpose of this paper is to obtain the small value probability of a supercritical branching process with immigration（Zn,n≥0） under the most relaxed condition,while the mean of offspring is finite but the L log L moment is infinite.It is well known that there exists a sequence of increasing constants {Cn,n≥0} such that Cn-1Zn converges to a finite non-degenerate limit W when the immigration distribution satisfies the log moment condition.Based on Chu,et al.＇s earlier work under a stronger condition about the offspring distribution,the convergence rate of P（W≤ε） as ε→0＋ is obtained.