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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Bibliographic Details
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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Summary: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.
ISSN:1532-6349
1532-4214
DOI:10.1080/15326349.2022.2066131