Skip to main content
SpringerLink
Log in

Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We consider a branching random walk on \({\mathbb {R}}\) with a stationary and ergodic environment \(\xi =(\xi _n)\) indexed by time \(n\in {\mathbb {N}}\). Let \(Z_n\) be the counting measure of particles of generation n and \(\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)\) be its Laplace transform. We show the \(L^p\) convergence rate and the uniform convergence of the martingale \(\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]\), and establish a moderate deviation principle for the measures \(Z_n\).

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. Alsmeyer, G., Iksanov, A., Polotsky, S., Rösler, U.: Exponential rate of \(L_p\)-convergence of intrinsic martingales in supercritical branching random walks. Theory Stoch. Process 15, 1–18 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Athreya, K.B., Karlin, S.: On branching processes in random environments I & II. Ann. Math. Stat. 42, 1499–1520 & 1843–1858 (1971)

  3. Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  4. Attia, N.: On the multifractal analysis of the branching random walk in \(R^d\). J. Theor. Probab. 27, 1329–1349 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25–37 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biggins, J.D.: Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14, 630–636 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Biggins, J.D.: Growth rates in the branching random walk. Z. Wahrsch. Verw. Geb. 48, 17–34 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biggins, J.D., Kyprianou, A.E.: Seneta–Heyde norming in the branching random walk. Ann. Probab. 25, 337–360 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Biggins, J.D.: The central limit theorem for the supercritical branching random walk, and related results. Stoch. Proc. Appl. 34, 255–274 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Biggins, J.D.: Uniform convergence of martingales in the one-dimensional branching random walk. Lecture Notes-Monograph Series, Vol. 18, Selected Proceedings of the Sheffield Symposium on Applied Probability (1991), pp. 159–173

  11. Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Probab. 20, 137–151 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Biggins, J.D., Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, 544–581 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chauvin, B., Rouault, A.: Boltzmann–Gibbs weights in the branching random walk. In: Athreya, K.B., Jagers, P. (eds.) Classical and Modern Branching Processes, IMA Vol. Math. Appl., vol. 84, pp. 41–50. Springer, New York (1997)

    Chapter  Google Scholar 

  14. Chow, Y.S., Teicher, H.: Probability Theory: Independence, Interchangeability and Martingales. Springer, New York (1988)

    Book  MATH  Google Scholar 

  15. Comets, F., Popov, S.: Shape and local growth for multidimensional branching random walks in random environment. ALEA Lat. Am. J. Probab. Math. Stat. 3, 273–299 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Comets, F., Yoshida, N.: Branching random walks in space-time random environment: survival probability, global and local growth rates. J. Theor. Probab. 24, 657–687 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dembo, A., Zeitouni, O.: Large deviations Techniques and Applications. Springer, New York (1998)

    Book  MATH  Google Scholar 

  18. Durrett, R., Liggett, T.: Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Geb. 64, 275–301 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gao, Z., Liu, Q., Wang, H.: Central limit theorems for a branching random walk with a random environment in time. Acta Math. Sci. Ser. B Engl. Ed. 34(2), 501–512 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Greven, A., den Hollander, F.: Branching random walk in random environment: phase transitions for local and global growth rates. Probab. Theory Relat. Fields 91, 195–249 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guivarc’h, Y.: Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré. Probab. Stat. 26, 261–285 (1990)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Hu, Y., Shi, Z.: Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37, 742–789 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang, C., Liu, Q.: Convergence in \(L^p\) and its exponential rate for a branching process in a random environment. Electron. J. Probab. 19(104), 1–22 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Huang, C., Liang, X., Liu, Q.: Branching random walks with random environments in time. Front. Math. China 9, 835–842 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huang, C., Liu, Q.: Branching random walk with a random environment in time. Available at arxiv:1407.7623

  27. Kahane, J.P., Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kaplan, N., Asmussen, S.: Branching random walks I & II. Stoch. Proc. Appl. 4, 1–13 & 15–31 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kuhlbusch, D.: On weighted branching processes in random environment. Stoch. Proc. Appl. 109, 113–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, Q.: Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten–Stigum concernant des processus de branchement. Adv. Appl. Probab. 29, 353–373 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, Q.: Fixed points of a generalized smoothing transformation and applications to branching processes. Adv. Appl. Probab. 30, 85–112 (1998)

    Article  MATH  Google Scholar 

  32. Liu, Q.: On generalized multiplicative cascades. Stoc. Proc. Appl. 86, 263–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu, Q.: Asymptotic properties absolute continuity of laws stable by random weighted mean. Stoch. Proc. Appl. 95, 83–107 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lyons, R.: A simple path to Biggins’s martingale convergence for branching random walk. In: Athreya, K.B., Jagers, P. (eds.) Classical and Modern Branching Processes, IMA Vol. Math. Appl., vol. 84, pp. 217–221. Springer, New York (1997)

    Chapter  Google Scholar 

  35. Nakashima, M.: Almost sure central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 21, 351–373 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yoshida, N.: Central limit theorem for random walk in random environment. Ann. Appl. Probab. 18, 1619–1635 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for valuable comments and remarks. This work has been partially supported by the National Natural Science Foundation of China (No. 11501146), and by the Shandong Provincial Natural Science Foundation of China (Nos. ZR2015PA003 and ZR2015AM017).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University (Weihai), Weihai, 264209, China

    Xiaoqiang Wang

  2. Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai, 264209, China

    Chunmao Huang

Authors
  1. Xiaoqiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunmao Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunmao Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Huang, C. Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time . J Theor Probab 30, 961–995 (2017). https://doi.org/10.1007/s10959-016-0668-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-016-0668-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation