Lower deviations for general supercritical branching process with immigration
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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．-
Key words:
- supercritical/
- immigration/
- branching process/
- lower deviations
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