Geodesic
Limit theorem for stationary distribution of a critical controlled branching process with immigration
Diskretnaya Matematika, Tome 35 (2023) no. 3, pp. 5-19.

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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. 
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V. I. Vinokurov. Limit theorem for stationary distribution of a critical controlled branching process with immigration. Diskretnaya Matematika, Tome 35 (2023) no. 3, pp. 5-19. http://geodesic.mathdoc.fr/item/DM_2023_35_3_a1/

[1] Sevastyanov B. A., Zubkov A. M., “Reguliruemye vetvyaschiesya protsessy”, Teoriya veroyatnostei i ee primeneniya, 19:1 (1974), 15–25 | Zbl

[2] Yanev N. M., “Usloviya vyrozhdeniya $\varphi$-vetvyaschikhsya protsessov so sluchainym $\varphi$”, Teoriya veroyatnostei i ee primeneniya, 20:2 (1975), 433–440 | MR | Zbl

[3] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P., Sluchainye razmescheniya, Nauka, M., 1976, 224 pp.

[4] Spravochnik po teorii veroyatnostei i matematicheskoi statistike, ed. Korolyuk V. S., Nauka, M., 1978, 582 pp. | MR