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

对于后代分布为{pi,i≥0}的上临界带移民分支过程{Zn},如果分支和移民分布满足适当的矩条件,则Zn/m^(n)几乎处处收敛到某个非退化的极限,其中m:=∑_(i=0)^(∞)ipi为过程后代分布的均值.本文给出了p0>0时该过程下偏差概率P(Zn=k)的渐近行为,其中k∈[k_(n),m^(n)],k_(n)→∞(n→∞),这一结果可作为文献[8]中Schroder情形结论的补充.

For a supercritical branching processes with immigration {Zn} with offspring distribution {pi,i≥0},it is known that under suitable conditions on the offspring and immigration distributions,Zn=mn 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 P(Zn=k)with k∈[k_(n),m^(n)],k_(n)→∞(n→∞).Detailed asymptotic behavior of such lower deviation probabilities is given as a complement to our previous work[8].