This paper studies the limit behaviour of   where Z_n is a real-valued temporally homogeneous Markov chain, and a_n and b_n are some constants; the results are then applied to a general population model. In such a model Z_n represents the nth generation population size and is defined as   are the offspring variables of the (n-1)th generation which are assumed to depend on n, i and Z_n-1 whereas the classical conditional independence of   given Z_n as superseded by milder assumptions. Some necessary and sufficient conditions for Z_n/b_n 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