Abstract:
We consider the sequence $\{{\xi_{n,t}}\}_{t\geq1} $ of controlled critical branching processes with immigration, where $n=1,2,\ldots$ is an integer parameter limiting the population size. It is shown that for $n\rightarrow\infty $ the stationary distributions of considered branching processes normalized by $\sqrt{n}$ converge to the distribution of a random variable whose square has a gamma distribution.
Keywords:controlled branching processes, Markov chain, stationary distribution, limit theorem, gamma distribution, the method of moments } In conclusion, the author expresses his sincere gratitude to A.M. Zubkov for his attention to the work and valuable comments. \begin{thebibliography}{99.