Limit theorems for a supercritical two-type decomposable branching process in a random environment

Let Z_n=(Z_n^(1)Z_n^(2) be a two-type decomposable branching process in an independent and identically distributed random environment, where a type 1 particle may produce particles of types 1 and 2, while a type 2 particle can only give birth to type 2 particles. We consider the asymptotic properties of this process in the supercritical case. Because Z_n^(1)is a usual single-type branching process in a random environment, we only consider Z_n^(2) First, under some moment 条件, we find a suitable factor Πn such that the normalized population size W_n=fracZ_n^(2)Πn