IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v82y2012i3p478-487.html
   My bibliography  Save this article

An estimate of the remainder of a limit theorem

Author

Listed:
  • He, Jianjun

Abstract

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with zero mean and finite variance. Set Sn=∑k=1nXk, EX2=σ2>0, λα(ϵ)=∑n=1∞P(|Sn|≥n1/2+αϵ), 0<α<1. In this paper, we discuss the rate of the approximation of σ1/αcα by ϵ1/αλα(ϵ) under suitable conditions, and extend the results of Klesov (1994), and He and Xie (in press), where cα=π−1/221/2αΓ(12+12α).

Suggested Citation

  • He, Jianjun, 2012. "An estimate of the remainder of a limit theorem," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 478-487.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:3:p:478-487
    DOI: 10.1016/j.spl.2011.11.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715211003622
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2011.11.009?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. Gut, Allan & Spataru, Aurel, 2003. "Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 398-422, August.
    2. Liu, Weidong & Lin, Zhengyan, 2006. "Precise asymptotics for a new kind of complete moment convergence," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1787-1799, October.
    3. Chen, Robert, 1978. "A remark on the tail probability of a distribution," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 328-333, June.
    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. Fa-mei Zheng & Qing-pei Zang, 2015. "A general pattern of asymptotic behavior of the R/S statistics for linear processes," Statistical Papers, Springer, vol. 56(1), pages 191-204, February.
    2. Xiao, Xiaoyong & Yin, Hongwei, 2012. "Precise asymptotics in the law of iterated logarithm for the first moment convergence of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1590-1596.
    3. A. Spătaru, 2004. "Exact Asymptotics in log log Laws for Random Fields," Journal of Theoretical Probability, Springer, vol. 17(4), pages 943-965, October.
    4. Aurel Spătaru, 1999. "Precise Asymptotics in Spitzer's Law of Large Numbers," Journal of Theoretical Probability, Springer, vol. 12(3), pages 811-819, July.
    5. Wei Huang, 2004. "Precise Rates in the Law of the Logarithm in the Hilbert Space," RePAd Working Paper Series lrsp-TRS396, Département des sciences administratives, UQO.
    6. Zhang, Li-Xin, 2021. "Heyde’s theorem under the sub-linear expectations," Statistics & Probability Letters, Elsevier, vol. 170(C).
    7. Yun-Xia, Li, 2006. "Precise asymptotics in complete moment convergence of moving-average processes," Statistics & Probability Letters, Elsevier, vol. 76(13), pages 1305-1315, July.
    8. Gut, Allan, 2002. "Precise asymptotics for record times and the associated counting process," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 233-239, October.
    9. Liu, Weidong & Lin, Zhengyan, 2006. "Precise asymptotics for a new kind of complete moment convergence," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1787-1799, October.
    10. Gut, Allan & Spataru, Aurel, 2003. "Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 398-422, August.
    11. Xie, Tingfan & He, Jianjun, 2012. "Rate of convergence in a theorem of Heyde," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1576-1582.
    12. Cai, Guang-hui & Wang, Jian-Feng, 2009. "Uniform bounds in normal approximation under negatively associated random fields," Statistics & Probability Letters, Elsevier, vol. 79(2), pages 215-222, January.
    13. Gut, Allan & Stadtmüller, Ulrich, 2011. "An intermediate Baum-Katz theorem," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1486-1492, October.
    14. Sergio Alvarez-Andrade & Salim Bouzebda, 2014. "Asymptotic results for hybrids of empirical and partial sums processes," Statistical Papers, Springer, vol. 55(4), pages 1121-1143, November.
    15. Kong, Lingtao & Dai, Hongshuai, 2016. "Convergence rate in precise asymptotics for Davis law of large numbers," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 295-300.

    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:82:y:2012:i:3:p:478-487. 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.