Skip to main content
SpringerLink
Log in

Limit theorems for a supercritical branching process with immigration in a random environment

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Afanasyev V I, Böinghoff C, Kersting G, et al. Limit theorems for weakly subcritical branching processes in random environment. J Theoret Probab, 2012, 25: 703–732

    Article  MathSciNet  MATH  Google Scholar 

  2. Afanasyev V I, Böinghoff C, Kersting G, et al. Conditional limit theorems for intermediately subcritical branching processes in random environment. Ann Inst H Poincaré Probab Statist, 2014, 50: 602–627

    Article  MathSciNet  MATH  Google Scholar 

  3. Athreya K B, Karlin S. On branching processes with random environments I: Extinction probabilities. Ann Math Statist, 1971, 42: 1499–1520

    Article  MathSciNet  MATH  Google Scholar 

  4. Athreya K B, Karlin S. On branching processes with random environments II: Limit theorems. Ann Math Statist, 1971, 42: 1843–1858

    Article  MathSciNet  MATH  Google Scholar 

  5. Athreya K B, Ney P E. Branching Processes. New York: Springer, 1972

    Book  MATH  Google Scholar 

  6. Bansaye V. Cell contamination and branching processes in a random environment with immigration. Adv Appl Probab, 2009, 41: 1059–1081

    MathSciNet  MATH  Google Scholar 

  7. Bansaye V, Berestycki J. Large deviations for branching processes in random environment. Markov Process Related Fields, 2009, 15: 493–524

    MathSciNet  MATH  Google Scholar 

  8. Bansaye V, Böinghoff C. Upper large deviations for branching processes in random environment with heavy tails. Electron J Probab, 2011, 16: 1900–1933

    Article  MathSciNet  MATH  Google Scholar 

  9. Bansaye V, Böinghoff C. Lower large deviations for supercritical branching processes in random environment. Proc Steklov Inst Math, 2013, 282: 15–34

    Article  MathSciNet  MATH  Google Scholar 

  10. Bansaye V, Böinghoff C. Small positive values for supercritical branching processes in random environment. Ann Inst H Poincaré Probab Statist, 2014, 50: 770–805

    Article  MathSciNet  MATH  Google Scholar 

  11. Bansaye V, Pardo J C, Smadi C. On the extinction of continuous state branching processes with catastrophes. Electron J Probab, 2013, 18: 1–31

    MathSciNet  MATH  Google Scholar 

  12. Barczy M, Li Z, Pap G. Moment formulas for multitype continuous state and continuous time branching process with immigration. J Theoret Probab, 2016, 29: 958–995

    Article  MathSciNet  MATH  Google Scholar 

  13. Böinghoff C, Kersting G. Upper large deviations of branching processes in a random environment—offspring distributions with geometrically bounded tails. Stochastic Process Appl, 2010, 120: 2064–2077

    Article  MathSciNet  MATH  Google Scholar 

  14. Chu W, Li W, Ren Y. Small value probabilities for continuous state branching processes with immigration. Sci China Math, 2012, 55: 2259–2271

    Article  MathSciNet  MATH  Google Scholar 

  15. Dawson D A. Measure-Valued Markov Processes. Berlin: Springer, 1993

    Book  MATH  Google Scholar 

  16. Dawson D A, Fleischmann K. A continuous super-Brownian motion in a super-Brownian medium. J Theoret Probab, 1997, 10: 213–276

    Article  MathSciNet  MATH  Google Scholar 

  17. Dawson D A, Fleischmann K, Le Gall J F. Super-Brownian motions in catalytic media. In: Branching Processses. Proceedings of the First World Congress. Lecture Notes in Statistics, vol. 99. New York: Springer-Verleg, 1999, 122–134

    Google Scholar 

  18. Dembo A, Zeitouni O. Large Deviations Techniques and Applications. New York: Springer, 1998

    Book  MATH  Google Scholar 

  19. Durrett R. Probability: Theory and Examples. Belmont: Duxbury Press, 2005

    MATH  Google Scholar 

  20. Grama I, Liu Q, Miqueu E. Asymptotic of the distribution and harmonic moments for a supercritical branching process in a random environment. ArXiv:1606.04228, 2016

    MATH  Google Scholar 

  21. Grama I, Liu Q, Miqueu E. Berry-Esseen bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment. Stochastic Process Appl, 2017, 127: 1255–1281

    Article  MathSciNet  MATH  Google Scholar 

  22. Grintsevichyus A K. The continuity of the distribution of a certain sum of dependent variables that is connected with independent walks on lines. Theory Probab Appl, 1974, 19: 163–168

    Article  MathSciNet  Google Scholar 

  23. Guivarc’h Y, Liu Q. Propriétés asympotiques des processus de branchement en environnement aléatoire. C R Acad Sci Paris Ser I, 2001, 332: 339–344

    Article  MATH  Google Scholar 

  24. He H, Li Z, Xu W. Continuous-state branching processes in Lévy random environments. J Theoret Probab, 2016, in press

    Google Scholar 

  25. Hong W, Li Z. A central limit theorem for super-Brownian motion with super-Brownian immigration. J Appl Probab, 1999, 36: 1218–1224

    Article  MathSciNet  MATH  Google Scholar 

  26. Hong W, Wang Z. Immigration process in catalytic medium. Sci China Ser A, 2000, 43: 59–64

    Article  MathSciNet  MATH  Google Scholar 

  27. Hong W, Zeitouni O. A quenched CLT for super-Brownian motion with random immigration. J Theoret Probab, 2007, 20: 807–820

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang C, Liu Q. Moment, moderate and large deviations for a branching process in a random environment. Stochastic Process Appl, 2012, 122: 522–545

    Article  MathSciNet  MATH  Google Scholar 

  29. Huang C, Liu Q. Convergence in Lp and its exponential rate for a branching process in a random environment. Electron J Probab, 2014, 19: 1–22

    Article  Google Scholar 

  30. Kesten H, Kozlov M V, Spitzer F. A limit law for random walk in a random environment. Compos Math, 1975, 30: 145–168

    MathSciNet  MATH  Google Scholar 

  31. Key E S. Limiting distributions and regeneration times for multitype branching processes with immigration in a random environment. Ann Probab, 1987, 15: 344–353

    Article  MathSciNet  MATH  Google Scholar 

  32. Le Gall J F. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Basel: Birkhäuser Verlag, 1999

    Book  MATH  Google Scholar 

  33. Li Y, Liu Q. Age-dependent branching processes in random environments. Sci China Ser A, 2008, 51: 1807–1830

    Article  MathSciNet  MATH  Google Scholar 

  34. Li Z. Branching processes with immigration and related topics. Front Math China, 2006, 1: 73–97

    Article  MathSciNet  MATH  Google Scholar 

  35. Li Z. Measure-Valued Branching Markov Processes. Berlin-Heidelberg: Springer, 2011

    Google Scholar 

  36. Liang X, Liu Q. Weighted moments of the limit of a branching process in a random environment. Proc Steklov Inst Math, 2013, 282: 127–145

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu Q, Rösler U. On the weighted branching process with immigration. Http://web.univ-ubs.fr/lmba/, 2016

    Google Scholar 

  38. Palau S, Pardo J C. Continuous state branching processes in random environment: The Brownian case. Stochastic Process Appl, 2017, 127: 957–994

    Article  MathSciNet  MATH  Google Scholar 

  39. Roitershtein A. A note on multitype branching processes with immigration in a random environment. Ann Probab, 2007, 35: 1573–1592

    Article  MathSciNet  MATH  Google Scholar 

  40. Smith W L, Wilkinson W. On branching processes in random environments. Ann Math Statist, 1969, 40: 814–827

    Article  MathSciNet  MATH  Google Scholar 

  41. Tanny D. A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stochastic Process Appl, 1988, 28: 123–139

    Article  MathSciNet  MATH  Google Scholar 

  42. Vatutin V A. A refinement of limit theorems for the critical branching processes in random environment. In: Workshop on Branching Processes and their Applications. Lecture Notes in Statistics, vol. 197. Berlin-Heidelberg: Springer, 2010, 3–19

    Chapter  Google Scholar 

  43. Vatutin V A. Multitype branching processes with immigration that evolve in a random environment, and polling systems (in Russian). Mat Tr, 2011, 14: 3–49

    MathSciNet  MATH  Google Scholar 

  44. Vatutin V A, Zheng X. Subcritical branching processes in a random environment without the Cramer condition. Stochastic Process Appl, 2012, 122: 2594–2609

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang M. Large deviation for super-Brownian motion with immigration. J Appl Probab, 2004, 41: 187–201

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang M. Moderate deviations for super-Brownian motion with immigration. Sci China Ser A, 2004, 47: 440–452

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China

    YanQing Wang

  2. Laboratoire de Mathématiques de Bretagne-Atlantique, Université de Bretagne-Sud, Vannes, F-56017, France

    QuanSheng Liu

Authors
  1. YanQing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. QuanSheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YanQing Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-016-9017-7

Keywords

MSC(2010)

Search

Navigation