Diffusion approximation of controlled branching processes using limit theorems for random step processes. (English) Zbl 1506.60092
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 T. N. Sriram et al. [Stochastic Processes Appl. 117, No. 7, 928–946 (2007; Zbl 1114.62082)]. An alternative proof is now provided making use of limit theorems for random step processes.
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
Keywords:
controlled branching processes; diffusion processes; martingale differences; random step processes; stochastic differential equation; weak convergence theoremCitations:
Zbl 1114.62082References:
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