Abstract
In this paper, a critical Galton-Watson branching process with immigration Zn is studied. We first obtain the convergence rate of the harmonic moment of Zn. Then the large deviation of \({S_{{Z_n}}}≔ \sum\nolimits_{i = 1}^{{Z_n}} {{X_i}} \) is obtained, where {Xi} is a sequence of independent and identically distributed zero-mean random variables with the tail index α > 2. We shall see that the converging rate is determined by the immigration mean, the variance of reproducing and the tail index of X +1 , compared with the previous result for the supercritical case, where the rate depends on the Schröder constant and the tail index.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11871103). The authors are deeply grateful to the anonymous referees for their careful reading and helpful suggestions to improve the paper.
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Li, D., Zhang, M. Harmonic moments and large deviations for a critical Galton-Watson process with immigration. Sci. China Math. 64, 1885–1904 (2021). https://doi.org/10.1007/s11425-019-1676-x
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DOI: https://doi.org/10.1007/s11425-019-1676-x