IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v147y2019icp18-28.html
   My bibliography  Save this article

A note on large deviation probabilities for empirical distribution of branching random walks

Author

Listed:
  • Shi, Wanlin

Abstract

We consider a branching random walk on R started from the origin. Let Zn(⋅) be the counting measure which counts the number of individuals at the nth generation located in a given set. For any interval A⊂R, it is well known that Zn(nA)Zn(R) converges a.s. to ν(A) under some mild conditions, where ν is the standard Gaussian measure. In this note, we study the convergence rate of PZ̄nnσ2A−ν(A)≥Δ,for a small constant Δ∈(0,1−ν(A)). Our work completes the results in Chen and He (2017) and Louidor and Perkins (2015), where the step size of the underlying walk is assumed to have Weibull tail, Gumbel tail or be bounded.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:147:y:2019:i:c:p:18-28
    DOI: 10.1016/j.spl.2018.11.029
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715218303845
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2018.11.029?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. Kaplan, Norman & Asmussen, Soren, 1976. "Branching random walks II," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 15-31, January.
    2. Asmussen, Soren & Kaplan, Norman, 1976. "Branching random walks I," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 1-13, January.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    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. Haojie Hou & Yan-Xia Ren & Renming Song, 2024. "Asymptotic Expansions for Additive Measures of Branching Brownian Motions," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3355-3394, November.
    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. 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.
    5. 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.
    6. 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.
    7. Huang, Chunmao & Liu, Quansheng, 2024. "Limit theorems for a branching random walk in a random or varying environment," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    8. 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.
    9. Durrett, R. & Lanchier, N., 2008. "Coexistence in host-pathogen systems," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 1004-1021, June.
    10. 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.
    11. Biggins, J. D. & Cohn, H. & Nerman, O., 1999. "Multi-type branching in varying environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 357-400, October.
    12. Bertoin, Jean, 2006. "Different aspects of a random fragmentation model," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 345-369, March.
    13. Johnson, Torrey, 2014. "On the support of the simple branching random walk," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 107-109.

    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:stapro:v:147:y:2019:i:c:p:18-28. 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/622892/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.