IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v142y2021icp293-318.html
   My bibliography  Save this article

Berry–Esseen bounds and moderate deviations for random walks on GLd(R)

Author

Listed:
  • Xiao, Hui
  • Grama, Ion
  • Liu, Quansheng

Abstract

Let (gn)n⩾1 be a sequence of independent and identically distributed random elements of the general linear group GLd(R), with law μ. Consider the random walk Gn:=gn…g1. Denote respectively by ‖Gn‖ and ρ(Gn) the operator norm and the spectral radius of Gn. For log‖Gn‖ and logρ(Gn), we prove moderate deviation principles under exponential moment and strong irreducibility conditions on μ; we also establish moderate deviation expansions in the normal range [0,o(n1/6)] and Berry–Esseen bounds under the additional proximality condition on μ. Similar results are found for the couples (Xnx,log‖Gn‖) and (Xnx,logρ(Gn)) with target functions, where Xnx:=Gn⋅x is a Markov chain and x is a starting point on the projective space P(Rd).

Suggested Citation

  • Xiao, Hui & Grama, Ion & Liu, Quansheng, 2021. "Berry–Esseen bounds and moderate deviations for random walks on GLd(R)," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 293-318.
    • Handle: RePEc:eee:spapps:v:142:y:2021:i:c:p:293-318
    DOI: 10.1016/j.spa.2021.08.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414921001290
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2021.08.005?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. Xiao, Hui & Grama, Ion & Liu, Quansheng, 2020. "Precise large deviation asymptotics for products of random matrices," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5213-5242.
    2. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.
    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. 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. Grama, Ion & Liu, Quansheng & Miqueu, Eric, 2017. "Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1255-1281.
    3. Gao, Zhenlong & Wang, Weigang, 2015. "Large deviations for a Poisson random indexed branching process," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 143-148.
    4. Xiao, Hui & Grama, Ion & Liu, Quansheng, 2023. "Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 103-133.
    5. Struleva, M.A. & Prokopenko, E.I., 2022. "Integro-local limit theorems for supercritical branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 181(C).
    6. Wang, Yuejiao & Liu, Zaiming & Li, Yingqiu & Liu, Quansheng, 2017. "On the concept of subcriticality and criticality and a ratio theorem for a branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 97-103.
    7. Gao, Zhenlong & Zhang, Yanhua, 2015. "Large and moderate deviations for a class of renewal random indexed branching process," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 1-5.
    8. Doukhan, Paul & Fan, Xiequan & Gao, Zhi-Qiang, 2023. "Cramér moderate deviations for a supercritical Galton–Watson process," Statistics & Probability Letters, Elsevier, vol. 192(C).
    9. Peter Eichelsbacher & Matthias Löwe, 2019. "Lindeberg’s Method for Moderate Deviations and Random Summation," Journal of Theoretical Probability, Springer, vol. 32(2), pages 872-897, June.
    10. Gao, Zhi-Qiang, 2021. "Exact convergence rate in the central limit theorem for a branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 178(C).

    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:142:y:2021:i:c:p:293-318. 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.