Fluctuation Limits of Controlled Branching Processes

M. González Velasco, I. M. del Puerto García, P. Martín-Chávez

This work deals with Controlled Multitype Branching Processes (CMBPs). These are stochastic processes used to model the evolution of populations with different types of individuals, where the number of progenitors of each type at a given generation is determined by a random control mechanism and the number of individuals of different types in the previous generation. We present a scaling limit theorem which establishes the asymptotic behavior of some critical CMBPs. It is proved that a sequence of appropriately scaled and normalized critical CMBPs converges weakly towards a certain squared Bessel process. This result extends the work by González et al. [Stoch. Models, 39, 1, 232-248 (2023)] to the multi-type case. Particular cases in the context of mutitype branching processes with immigration and promiscuous two-sex Galton-Watson branching processes with immigration are also provided.

Keywords: controlled branching processes; weak convergence; scaling limits

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