# On central limit theorems for branching processes with dependent immigration

### 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

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*Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics*, (1-2), 7-15. https://doi.org/10.17721/1812-5409.2020/1-2.1

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