This paper studies the limit behaviour ofwhere {Zn} is a real-valued temporally homogeneous Markov chain, and {an} and {bn} are some constants; the results are then applied to a general population model. In such a model Znrepresents the nth generation population size and is defined asare the offspring variables of the (n-1)th generation which are assumed to depend on n, i and Zn-1whereas the classical conditional independence ofgiven Znas superseded by milder assumptions. Some necessary and sufficient conditions for {Zn/bn} to converge a.s. are derived, and some results on the robustness of the asymptotic behaviour of the Galton-Watson process are obtained when offspring independence is relaxed