一类上临界带移民分枝过程的下偏差估计

Lower deviations for general supercritical branching process with immigration

摘要: 对于一类带移民的上临界分枝过程 (Z_n) ，存在一列正常数 c_n 可以用来描述过程的增长速度．任取一列满足 k_n\rightarrow \infty 和 k_n=o( c_n) 的正常数 k_n ， P(Z_n=k_n) 的渐近行为即为 Z_n 的下偏差．假设 EZ_1 \ln Z_1=\infty ：1）证明了过程 Z_n 的一个局部极限定理；2）给出了在 Schröder和Böttcher情形下 Z_n 的下偏差估计，补充并完善了已有文献的结果．

Abstract: For a supercritical branching process with immigration (Z_n), a sequence of constant c_n could be used to describe the growth rate of the process．The asymptotic behavior of P(Z_n=k_n) (k_n=o(c_n)) is called the lower deviation probability of Z_n ．In this paper, under EZ_1 \ln Z_1=\infty , first, a local limit theorem of Z_n is proved．Then in the Schröder and Böttcher cases, the lower deviation probability P(Z_n=k_n) is discussed, which improves and generalizes the corresponding results in the literature．