4.0 Article

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

期刊

STOCHASTIC MODELS
卷 39, 期 1, 页码 232-248

出版社

TAYLOR & FRANCIS INC
DOI: 10.1080/15326349.2022.2066131

关键词

Controlled branching processes; diffusion processes; Martingale differences; random step processes; stochastic differential equation; weak convergence theorem

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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.
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.

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