Small value probabilities for supercritical branching processes with immigration

Abstract

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.