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

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.