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We consider the sequence { ξ   n , t  }  t ≥1  of controlled critical branching processes with immigration, where  n  = 1, 2, … is an integer parameter limiting the population size. It is shown that for  n  → ∞ the stationary distributions of considered branching processes normalized byn$\begin{array}{} \sqrt{n} \end{array} $    converge to the distribution of a random variable whose square has a gamma distribution.

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