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Harmonic moments and large deviations for supercritical branching processes with immigration

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Abstract

We study the convergence rates of the harmonic moments for supercritical branching processes with immigration Z n , extending the previous results for non-immigration cases in literature. As a by-product, the large deviations for Z n+1/Z n are also studied. We can see that there is a phase transition in converging rates depending on the generating functions of both branching and immigration.

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References

  1. Athreya K B. Large deviation rates for branching processesI. Single type case. Ann Appl Probab, 1994, 4(3): 779–790

    Article  MATH  Google Scholar 

  2. Athreya K B, Ney P E. Branching Processes. Berlin-New York: Springer-Verlag,1972

    Book  MATH  Google Scholar 

  3. Harris T E. Branching Processes. Ann Math Stat, 1948, 19(4): 474–494

    Article  MathSciNet  MATH  Google Scholar 

  4. Heyde C, Brown B. An invariance principle and some convergence rate results for branching processes. Z Wahrsch Verw Gebiete, 1971, 20(4): 271–278

    Article  MathSciNet  MATH  Google Scholar 

  5. Lin Z, Bai Z. Probability Inequalities. Beijing: Science Press,2010 (in Chinese)

    MATH  Google Scholar 

  6. Liu J, Zhang M. Large deviation for supercritical branching processes with immigration. Acta Math Sin (Engl Ser), 2016, 32(8): 893–900

    Article  MathSciNet  MATH  Google Scholar 

  7. Nagaev A V. On estimating the expected number of direct descendants of a particle in a branching process. Theory Probab Appl, 1967, 12(2): 314–320

    Article  MATH  Google Scholar 

  8. Ney P E, Vidyashankar A N. Harmonic moments and large deviation rates for super-critical branching processes. Ann Appl Probab, 2003, 13(2): 475–489

    Article  MathSciNet  MATH  Google Scholar 

  9. Pakes A G. Non-parametric estimation in the Galton-Watson process. Math Biosci, 1975, 26: 1–18

    Article  MathSciNet  MATH  Google Scholar 

  10. Royden H L. Real Analysis. 2nd ed. New York: The Macmillan,1968

    MATH  Google Scholar 

  11. Seneta E. On the supercritical Galton-Watson process with immigration. Math Biosci, 1970, 7: 9–14

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11371061).

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Authors and Affiliations

  1. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, China

    Qi Sun & Mei Zhang

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  1. Qi Sun
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  2. Mei Zhang
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Correspondence to Mei Zhang.

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Sun, Q., Zhang, M. Harmonic moments and large deviations for supercritical branching processes with immigration. Front. Math. China 12, 1201–1220 (2017). https://doi.org/10.1007/s11464-017-0642-3

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  • DOI: https://doi.org/10.1007/s11464-017-0642-3

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