Small Positive Values and Limit Theorems for Supercritical Branching Processes with Immigration in Random Environment

Abstract

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_n 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.