Popoulation-size-dependent branching processes

In a recent paper a coupling method was used to show that if population size, or more generally population history, influence upon individual reproduction in growing, branching-style populations disappears after some random time, then the classicl Malthusian properties of exponential growth and stabilization of composition persist. While this seems self-evident, as stated, it is interesting that it leads to neat criteria via a direct Borel-Cantelli argument: If is the expected number of children of an individual in an n-size population and , then essentially suffices to guarantee Malthusian behavior with the same parameter as a limiting independent-individual process with expected offspring number m. (For simplicity the criterion is stated for the single-type case here.) However, this is not as strong as the results known for the special cases of Galton-Watson process, Markov branching, and a binary splitting tumor model, which all require only something like . This note studies such latter criteria more generally.