Geometric rate of growth in markov chains with applications to population-size-dependent models with dependent offspring

This paper studies the limit behaviour of {(Z_n − a_n)/b_n} 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 [formulla omitted] where {X_i ^n-1} 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assumed to depend on n, i and Z_n−1’ whereas the classical conditional independence of {x_i ⁿ, i=1,…,Z_n} given Z_n is superseded by milder assumptions.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.