# On central limit theorems for branching processes with dependent immigration

## DOI:

https://doi.org/10.17721/1812-5409.2020/1-2.1## Abstract

In this paper we consider subcritical and supercritical discrete time branching processes with generati-on dependent immigration. We prove central limit theorems for fluctuation of branching processes with immigration when the mean of immigrating individuals tends to infinity with the generation number and immigration process is m?dependent. The first result states on weak convergence of the fluctuati-on subcritical branching processes with m?dependent immigration to standard normal distribution. In this case, we do not assume that the mean and variance of immigration process are regularly varying at infinity. In contrast, in Theorem 3.2, we suppose that the mean and variance are to be regularly varying at infinity. The proofs are based on direct analytic method of probability theory.

* Key words*: Branching process, immigration, m?dependence.

* Pages of the article in the issue*: 7 - 15

** Language of the article**: English

## References

Asadullin, M.Kh. and Nagaev, S.V. (1982) Limit theorems for critical branching processes with immigration, Math Notes, Vol. 32, pp. 537-548.

Asmussen, S. and Hering, H. (1983) "Branching processes", Birkhauser, Boston.

Athreya, K.B. and Ney, P.E. (1972) "Branching processes", Springer-Verlag, New York.

Badalbaev, I.S. and Rahimov, I. (1978) "Critical branching processes with immigration of decreasing intensity", Theory of Probabili-ty and its Applications, Vol. 23, pp. 275-283.

Badalbaev, I.S. and Zubkov, A.M. (1983) "A limit theorem for sequence of branching processes with immigration", Theory Probability and its Applications, Vol. 28, pp. 382-388.

Diananda, P.H. (1955) "The central limit theorem for m-dependent random variables", Proc. Comb. Phil. Soc., Vol. 51, pp. 192-195.

Foster, J.H. and Williamson, J.A. (1971) " Limit theorems for the Galton-Watson process with time-dependent immigration", Z. Wahrschein. Und Verw. Ceb., Vol. 20, pp. 227-235.

Galambos, J. and "Regularly varying ngs of the American Vol. 41, pp. 110-116.

Seneta, E. (1973), sequences", Proceedi-Mathematical Society,

Guo, H. and Zhang, M. (2014), "A fluctuation limit theorem for a critical branching process with dependent immigration", Statistics and Probability Letters, Vol. 94, pp. 29-38.

Heathcote, C.R. (1965), "A branching process allowing immigration", J. R. Statist. Soc., Vol. 27, pp. 138-143.

Hering, H. (2014), "Asymptotic behaviour of immigration-branching processes with general set of types. I: Critical branching part", Adv. Appl. Prob., Vol. 5, pp. 391-416.

Heyde, C.C. and Seneta, E. (1971), "Analogues of classical limit theorems for the super- critical Galton-Watson process with immigration", Math. Biosci., Vol. 11, pp. 249-259.

Iksanov, A. and Kabluchko, Z. (2018), "Functional limit theorems for Galton-Watson processes with very active immigration", Stochastic processes and their applications, Vol. 128, pp. 291-305.

Ispainy, M. and Pap, G. and Van Zuijlen, M.C.A. (2005), "Fluctuation limit theorem of branching processes with immigration and estimation of the mean", Adv.Appl.Probab., Vol. 37, pp. 523-538.

Ispainy, M. (2008), "Limit theorems for normalized nearly critical branching processes with immigration", Publ. Math.Decreben, Vol. 72, pp. 17-34. 16. Khusanbaev, Ya.M. (2010), "On the asymptotics of a critical branching process with heterogeneous and increasing immigration", Doklady of Academy of Sciences RUz, pp. 6-10.

Khusanbaev, Ya.M. (2013), "On the asymptotic behavior of a subcritical branching process with immigration", Ukrainian Mathematical Journal, Vol. 65, pp. 835-843.

Khusanbaev, Ya.M. (2014), "On the convergence rate in one limit theorem for branching processes with immigration", Siberian Mathematical Journal, Vol. 55, pp. 178-187.

Khusanbaev, Ya.M. (2016), "On asymptotics of branching processes with immigration", Discrete Math.Appl., Vol. 28, pp. 113-122.

Khusanbaev, Ya. and Sharipov, S. and Golomoziy, V. (2019), "Berry-Esseen bound for nearly critical branching processes with immigration", Bulletin of Taras Shevchenko National University of Kyiv Series: Physics and Mathematics, Vol. 4, pp. 42-49.

Khusanbaev, Ya.M. and Sharipov, S.O. (2020), "On branching the asymptotic behavior of processes with stationary immigration", Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, Vol. 3, pp. 59-73.

Khusanbaev, Y.M. and Sharipov, S.O. (2017), "Functional limit theorem for critical branching processes with dependent immigration", Uzbek Mathematical Journal, Vol 3, pp. 149-158.

Khusanbaev, Y.M. and Sharipov, S.O. (2018), "On convergence of branching processes with weakly dependent immigration", Uzbek Mathematical Journal, Vol 1, pp. 108-114.

Li, Z.H. (2000), "Ornstein-Uhlenbeck type processes and branching processes with immigration", J. Appl. Probab., Vol. 37, pp. 627-634.

Li, Z.H. (2006), "Branching processes with immigration and related topics.", Frontiers of Mathematics in China, Vol. 1, pp. 73-97.

Mitov, K. V. and Yanev, N. M. (2006), "Critical branching processes with decreasing state-dependent immigration", Acad. Bulgar. Sci., Vol. 36, pp. 193-196.

Mitov, K. V. and Yanev, N. M. (1984), "Critical Galton-Watson processes with decreasing state-dependent immigration", J. Appl. Prob., Vol. 21, pp. 22-39.

Mitov, K. V. and Yanev, N. M. (1984), "Branching processes with decreasing state-dependent immigration", Serdica, Vol. 10, pp. 13-21.

Nagaev, S.V. (1975), "Limit theorem for branching processes with immigration", Theory Probab. Appl., Vol. 20, pp. 178-180.

Pakes, A.G. (1971), "Some results for the supercritical branching process with immigration", Math. Biosciences, Vol 11, pp. 355-363.

Pakes, A.G. (1971), "A branching process with a state dependent immigration component", Advances in Applied Probability, Vol. 3, pp. 301-314.

Pakes, A.G. (1975), "Some results for non-supercritical Galton-Watson processes with immigration", Math. Biosciences, Vol. 24, pp. 71-92.

Rahimov, I. (1978), "On critical Galton-Watson process with increasing immigration", Izvestiya AN UZSSR. Seriya fiz.-mat.nauk, Vol. 4, pp. 22-27.

Rahimov, I. (1981), "On branching random processes with increasing immigration", Doklady AN UZSSR, Vol. 1, pp. 3-5.

Rahimov, I. (1995), "Random Sums and Branching Stochastic Processes", Springer, New York.

Rahimov, I. (2007), "Functional limit theorems for critical processes with immigration", Advances in Applied Probability, Vol. 39, pp. 1054-1069.

Rahimov, I. (2008), "Deterministic approximation of a sequence of nearly critical branching processes", Stochastic Analysis and Applications, Vol. 26, pp. 1013-1024.

Rahimov, I. (2012), "Conditional least squares estimators for the offspring mean in a subcri-tical process with immigration", Stochastic Analysis and Applications, Vol. 41, pp. 2096-2110.

Rahimov, I. and Sirazhdinov, S.K. (1988), "Approximation of the distribution of a sum in a scheme for the summation of independent random variables", Dokl. Math., Vol. 38, pp. 23-27.

Seneta, E. (1968), "On asymptotic properties of subcritical branching processes", J. Australian Math. Soc., Vol. 8, pp. 671-682.

Seneta, E. (1970), "An explicit-limit theorem for the Critical Galton-Watson process with immigration", Journal of the Royal Statistical Society, Vol. 32, pp. 149-152.

Sevast'yanov, B.A. (1957), "Limit theorems for branching processes of special form", Theory Probab. Appl., Vol. 2, pp. 339-348.

Sriram, T.N. (1994), "Invalidity of bootstrap for critical branching processes with immigration", Ann.Statist., Vol. 22, pp. 1013-1023.

Shiryaev, A.N. (2004), "Probability", MCNMO, Moscow.

Vatutin, V.A. and Zubkov, A.M. (1985), "Branching processes. r', Itogi Nauki i Tekhniki. Ser. Teor. Veroyatn. Mat. Stat. Teor. Kibern., Vol. 23, pp. 3-67.

Wei, C.Z. and Winnicki, J. (1989), "Some asymptotic results for the branching process with immigration", Stochastic processes their applications, Vol. 31, pp. 261-282.

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (1-2), 7–15. https://doi.org/10.17721/1812-5409.2020/1-2.1

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