Galton-Watson theta-processes in a varying environment

We consider a special class of Galton-Watson theta-processes in a varying environment fully defined by four parameters, with two of them $(\theta ,r)$ being fixed over time $n$, and the other two $(a_n,c_n)$ characterizing the altering reproduction laws. We establish a sequence of transparent limit theorems for the theta-processes with possibly defective reproduction laws. These results may serve as a stepping stone towards incisive general results for the Galton-Watson processes in a varying environment.