On a method of proving limit theorems for branching processes with immigration

Previously, S. V. Nagaev and the author obtained a result concerning a scheme of summation of a random number of independent random variables and suggested a new method of proving limit theorems for branching processes with immigration. This method provides a possibility of obtaining asymptotic results on branching processes with immigration from the corresponding results for the branching processes without immigration. In this paper we consider in detail the supercritical and subcritical cases which were not yet considered.