Critical Galton–Watson Processes with Overlapping Generations

Abstract 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\geq 0 ) with a pertinent \gamma\geq 0 .