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

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...

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Published in:Stochastic models 2023-01, Vol.39 (1), p.232-248
Main Authors: González, Miguel, Martín-Chávez, Pedro, del Puerto, Inés M.
Format: Article
Language:English
Subjects:
Approximation
Branching (mathematics)
Controlled branching processes
Diffusion
diffusion processes
Markov processes
Martingale differences
Mathematical analysis
Random numbers
random step processes
stochastic differential equation
Stochastic processes
Theorems
weak convergence theorem
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description 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.
doi_str_mv 10.1080/15326349.2022.2066131
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subjects Approximation
Branching (mathematics)
Controlled branching processes
Diffusion
diffusion processes
Markov processes
Martingale differences
Mathematical analysis
Random numbers
random step processes
stochastic differential equation
Stochastic processes
Theorems
weak convergence theorem
title Diffusion approximation of controlled branching processes using limit theorems for random step processes
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