Diffusion approximation of controlled branching processes using limit theorems for random step processes

Abstract A controlled branching process (CBP) is a modification of the standard Bienaymé–Galton–Watson process in which the number of progenitors in each generation is determined by a random mechanism. We consider a CBP starting from a random number of initial individuals. The main aim of this article is to provide a Feller diffusion approximation for critical CBPs. A similar result by considering a fixed number of initial individuals by using operator semigroup convergence theorems has been previously proved by Sriram et al. (Stochastic Processes Appl. 2007;117:928–946). An alternative proof is now provided making use of limit theorems for random step processes.