IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v126y2016i9p2634-2664.html
   My bibliography  Save this article

Exact convergence rates in central limit theorems for a branching random walk with a random environment in time

Author

Listed:
  • Gao, Zhiqiang
  • Liu, Quansheng

Abstract

Chen (2001) derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process. We extend Chen’s results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen’s by relaxing the second moment condition used by Chen to a moment condition of the form EX(ln+X)1+λ<∞. In the rate functions that we find for a branching random walk, we figure out some new terms which did not appear in Chen’s work. The results are established in the more general framework, i.e. for a branching random walk with a random environment in time. The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment.

Suggested Citation

  • Gao, Zhiqiang & Liu, Quansheng, 2016. "Exact convergence rates in central limit theorems for a branching random walk with a random environment in time," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2634-2664.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:9:p:2634-2664
    DOI: 10.1016/j.spa.2016.02.013
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414916000430
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2016.02.013?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hu, Yueyun & Yoshida, Nobuo, 2009. "Localization for branching random walks in random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1632-1651, May.
    2. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    3. Kaplan, Norman & Asmussen, Soren, 1976. "Branching random walks II," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 15-31, January.
    4. Asmussen, Soren & Kaplan, Norman, 1976. "Branching random walks I," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 1-13, January.
    5. Biggins, J. D., 1990. "The central limit theorem for the supercritical branching random walk, and related results," Stochastic Processes and their Applications, Elsevier, vol. 34(2), pages 255-274, April.
    6. Tanny, David, 1988. "A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means," Stochastic Processes and their Applications, Elsevier, vol. 28(1), pages 123-139, April.
    7. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    2. Deng, Guantie & Fan, Xiequan & Gao, Zhi-Qiang, 2023. "Asymptotic expansions in the local limit theorem for a branching Wiener process," Statistics & Probability Letters, Elsevier, vol. 199(C).
    3. Gao, Zhi-Qiang, 2019. "Exact convergence rate in the local central limit theorem for a lattice branching random walk on Zd," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 58-66.
    4. Gao, Zhiqiang, 2017. "Exact convergence rate of the local limit theorem for branching random walks on the integer lattice," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1282-1296.
    5. Gao, Zhi-Qiang, 2018. "A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4000-4017.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xiaoqiang Wang & Chunmao Huang, 2017. "Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 961-995, September.
    2. Gao, Zhi-Qiang, 2018. "A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4000-4017.
    3. Gao, Zhi-Qiang, 2019. "Exact convergence rate in the local central limit theorem for a lattice branching random walk on Zd," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 58-66.
    4. Gao, Zhiqiang, 2017. "Exact convergence rate of the local limit theorem for branching random walks on the integer lattice," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1282-1296.
    5. Shi, Wanlin, 2019. "A note on large deviation probabilities for empirical distribution of branching random walks," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 18-28.
    6. Durrett, R. & Lanchier, N., 2008. "Coexistence in host-pathogen systems," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 1004-1021, June.
    7. Vincent Bansaye, 2019. "Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment," Journal of Theoretical Probability, Springer, vol. 32(1), pages 249-281, March.
    8. Bertoin, Jean, 2006. "Different aspects of a random fragmentation model," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 345-369, March.
    9. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    10. Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    11. Alsmeyer, Gerold & Gröttrup, Sören, 2016. "Branching within branching: A model for host–parasite co-evolution," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1839-1883.
    12. Borovkov, K. & Motyer, A., 2005. "On the asymptotic behaviour of a simple growing point process model," Statistics & Probability Letters, Elsevier, vol. 72(3), pages 265-275, May.
    13. Olvera-Cravioto, Mariana, 2012. "Tail behavior of solutions of linear recursions on trees," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1777-1807.
    14. Francis Comets & Nobuo Yoshida, 2011. "Branching Random Walks in Space–Time Random Environment: Survival Probability, Global and Local Growth Rates," Journal of Theoretical Probability, Springer, vol. 24(3), pages 657-687, September.
    15. Decrouez, Geoffrey & Hambly, Ben & Jones, Owen Dafydd, 2015. "The Hausdorff spectrum of a class of multifractal processes," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1541-1568.
    16. Buraczewski, D. & Damek, E. & Zienkiewicz, J., 2018. "Pointwise estimates for first passage times of perpetuity sequences," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2923-2951.
    17. Iksanov, Aleksander M., 2004. "Elementary fixed points of the BRW smoothing transforms with infinite number of summands," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 27-50, November.
    18. Buraczewski, Dariusz & Damek, Ewa & Mentemeier, Sebastian & Mirek, Mariusz, 2013. "Heavy tailed solutions of multivariate smoothing transforms," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1947-1986.
    19. Liu, Quansheng & Watbled, Frédérique, 2009. "Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3101-3132, October.
    20. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:126:y:2016:i:9:p:2634-2664. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.