Lower Deviations for Supercritical Branching Processes with Immigration Concerning a Special Case

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.