Limit theorem for stationary distribution of a critical controlled branching process with immigration

Key Points

• The stationary distributions converge to a gamma-distributed random variable as the population size approaches infinity.
• Normalization by the square root of the population size demonstrates convergence behavior of the distributions.
• Analysis of controlled branching processes with immigration provides insights into population dynamics and stability.
• Further studies on critical controlled branching processes may uncover additional properties and behaviors related to immigration effects.