Population-size-dependent branching processes

In a recent paper [7] 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 classical 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 m ( n ) is the expected number of children of an individual in an n -size population and m ( n ) ≥ m > 1 , then essentially ∑ n = 1 ∞ { m ( n ) − m } < ∞ 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 processes [10], Markov branching [13], and a binary splitting tumor model [2], which all require only something like ∑ n = 1 ∞ { m ( n ) − m } / n < ∞ . This note studies such latter criteria more generally. It is dedicated to the memory of Roland L. Dobrushin.