A scaling limit theorem for controlled branching processes with a size-divisible term

2025/08/23 by González, Miguel, Martín-Chávez, Pedro, del Puerto, Inés
#60F05 #60J80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2508.17116

Abstract

We establish general sufficient conditions for a sequence of controlled branching processes to converge weakly on the Skorokhod space. We focus on a class of controlled random variables that extends previous results by considering them as a sum of an immigration size-dependent term and a size-divisible term. Our assumptions are established in terms of the probability generating functions of the offspring and control distributions, distinguishing in this latter case between the immigration and the size-divisible parts. The limit process is a continuous-state process with dependent immigration.

Citations

• Linking Population-Size-Dependent and Controlled Branching Processes
• Sample paths of continuous-state branching processes with dependent immigration
• Continuous-state branching processes with immigration
• Some asymptotic results for the branching process with immigration

Related

• Scaling Limits of Controlled Branching Processes
• Limit theorems for some branching measure-valued processes
• Weak convergence of continuous-state branching processes with large immigration
• Diffusion approximation of a sequence of critical branching processes with dependent immigration
• Continuous-state branching processes