Diffusion approximation of critical controlled multi-type branching processes
| Autor: | Barczy, Matyas, González, Miguel, Martín-Chávez, Pedro, del Puerto, Inés |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Revista de la Real Academia de Ciencias Exactas, F\'isicas y Naturales. Serie A. Matem\'aticas 118, (2024), Article Number 101, 36 pages |
| Druh dokumentu: | Working Paper |
| Popis: | Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs. Comment: 41 pages |
| Databáze: | arXiv |
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