Critical branching processes with nonhomogeneous migration (Q1064674)

A Galton-Watson branching process is modified in the following way: at generation n, there is probability \(p_ n\) that just one of the individuals present (if any) is eliminated before reproduction; probability \(r_ n\) that, in addition to reproduction, immigration occurs according to some specified (time-homogeneous) distribution with mean m; and probability \(q_ n\) that reproduction occurs in the standard way, where \(p_ n+q_ n+r_ n=1\). In the critical case with \(p_ n=mr_ n\) and \(p_ n\) converging to zero in regularly varying fashion, conditional limit theorems for the population size are obtained.