Small value probabilities for supercritical branching processes with immigration

We consider a supercritical Galton-Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit $\mathcal{W}$ for the normalized population size. Small value probabilities for $\mathcal{W}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized.