Critical Galton-Watson processes with overlapping generations

A properly scaled critical Galton-Watson process converges to a continuous state critical branching process $\xi(\cdot)$ as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals $(\int_0^y\xi(y-u)du^\gamma, y\ge0)$ with a pertinent $\gamma\ge0$.