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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References
Athreya K B, Ney P E. Branching Processes. Berlin: Springer, 1972
Athreya K B. Large deviation rates for branching processes I: Single type case. Ann Appl Probab, 1994, 4: 779–790
Borovkov A A. Estimates for sums and maxima of sums of random variables when the Cramér condition is not satisfied. Sib Math J, 2000, 41: 811–848
Fleischmann K, Wachtel V. Large deviations for sums indexed by the generations of a Galton-Watson process. Probab Theory Related Fields, 2008, 141: 445–470
Heyde C, Brown B. An invariance principle and some convergence rate results for branching processes. Z Wahrsch Verw Gebiete, 1971, 20: 271–278
Kallenberg O. Foundations of Modern Probability. New York: Springer, 2002
Kesten H, Ney P, Spitzer F. The Galton-Watson process with mean one and finite variance. Theory Probab Appl, 1966, 11: 579–611
Li D D, Zhang M. Asymptotic behaviors for critical branching processes with immigration. Acta Math Sin Engl Ser, 2019, 35: 537–549
Liu J N, Zhang M. Large deviation for supercritical branching processes with immigration. Acta Math Sin Engl Ser, 2016, 32: 893–900
Mellein B. Local limit theorems for the critical Galton-Watson process with immigration. Rev Colombiana Mat, 1982, 16: 31–56
Nagaev A V. On estimating the expected number of direct descendants of a particle in a branching process. Theory Probab Appl, 1967, 12: 314–320
Nagaev S V. Large deviations of sums of independent random variables. Ann Probab, 1979, 7: 745–789
Nagaev S V, Vachtel V I. On the local limit theorem for a critical Galton-Watson process. Theory Probab Appl, 2006, 50: 400–419
Ney P E, Vidyashankar A N. Harmonic moments and large deviation rates for supercritical branching processes. Ann Appl Probab, 2003, 13: 475–489
Ney P E, Vidyashankar A N. Local limit theory and large deviations for supercritical branching processes. Ann Appl Probab, 2004, 14: 1135–1166
Pakes A G. Further results on the critical Galton-Watson process with immigration. J Aust Math Soc, 1972, 13: 277–290
Pakes A G. Non-parametric estimation in the Galton-Watson processes. Math Biosci, 1975, 26: 1–18
Petrov V V. Sums of Independent Random Variables. Berlin: Springer-Verlag, 1975
Sun Q, Zhang M. Harmonic moments and large deviations for supercritical branching processes with immigration. Front Math China, 2017, 12: 1201–1220
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