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

A controlled branching process (CBP) is a modification of the standard Bienayme-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.

Automated summary

The article aims to provide a Feller diffusion approximation for critical controlled branching processes (CBPs). A similar result has been previously proved by considering a fixed number of initial individuals using operator semigroup convergence theorems. An alternative proof is now provided using limit theorems for random step processes.