Population-size-dependent branching processes

Abstract

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.