Abstract
Let (Zn) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size Wn converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Zn.
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Acknowledgments
This work was supported by National Natural Science Foundation of China (Grants Nos. 11401590 and 11571052). It has also beneted from a visit of Yanqing Wang to Laboratoire de Mathématiques de Bretagne-Atlantique, Université de Bretagne-Sud, and a visit of Quansheng Liu to the School of Statistics and Mathematics, Zhongnan University of Ecnomics and Law. The support and the hospitality of both universities have been well appreciated. The authors are grateful to two anonymous referees for their very valuable comments and remarks, which signicantly contributed to improving the quality of the paper.
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Wang, Y., Liu, Q. Limit theorems for a supercritical branching process with immigration in a random environment. Sci. China Math. 60, 2481–2502 (2017). https://doi.org/10.1007/s11425-016-9017-7
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DOI: https://doi.org/10.1007/s11425-016-9017-7
Keywords
- branching process with immigration
- random environment
- almost sure convergence
- nondegeneration
- L pconvergence and moments
- large and moderate deviations
- central limit theorem