一类上临界带移民分支过程的下偏差概率估计
Lower Deviations for Supercritical Branching Processes with Immigration Concerning a Special Case
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摘要: 对于后代分布为\p_i,i\ge 0\的上临界带移民分支过程\Z_n\, 如果分支和移民分布满足适当的矩条件,则Z_n/m^n几乎处处收敛到某个非退化的极限,其中m:=\sum_i=0^\infty ip_i为过程后代分布的均值.本文给出了p_0>0时该过程下偏差概率\pr(Z_n=k)的渐近行为,其中k\ink_n,m^n, k_n\to\infty (n\rightarrow\infty),这一结果可作为文献\ncite8中Schr\"oder情形结论的补充.Abstract: For a supercritical branching processes with immigration \Z_n\ with offspring distribution \p_i,i\ge 0\, it is known that under suitable conditions on the offspring and immigration distributions, Z_n/m^n converges almost surely to a finite and strictly positive limit, where m is the offspring mean. In certain situation p_0>0, we study the limiting properties of the probabilities \pr(Z_n=k) with k\ink_n,m^n, k_n\to\infty as n\rightarrow\infty. Detailed asymptotic behavior of such lower deviation probabilities is given as a complement to our previous work \ncite8.