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

TL;DR

This paper develops a diffusion approximation for critical controlled branching processes (CBPs) starting from a random initial population, using limit theorems for random step processes, providing an alternative to previous operator semigroup methods.

Contribution

It introduces a new proof for diffusion approximation of CBPs using limit theorems for random step processes, extending prior results to random initial populations.

Findings

Established Feller diffusion approximation for critical CBPs with random initial size.

Provided an alternative proof method using limit theorems for random step processes.

Extended previous results that used operator semigroup convergence.

Abstract

A controlled branching process (CBP) is a modification of the standard Bienaym\'e-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 paper 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 in Sriram at al. (2007). An alternative proof is now provided making use of limit theorems for {random step processes}.