Small positive values and limit theorems for supercritical branching processes with immigration in random environment

Let $(Z_n)$ be a supercritical branching process with immigration in a random environment. The small positive values and some lower deviation inequalities for $Z$ are investigated. Based on these results, the central limit theorem of $\log Z_n$ and the Edgeworth expansion are obtained. The study is taken under the assumption that each individual produces $0$ offspring with a positive probability.