On possible rates of growth of age-dependent branching processes with immigration

Abstract

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter. As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.