IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v61y2003i4p347-357.html
   My bibliography  Save this article

On strong versions of the central limit theorem

Author

Listed:
  • Rychlik, Zdzislaw
  • Szuster, Konrad S.

Abstract

The purpose of this paper is to present strong versions of the central limit theorem for independent nonidentically distributed random variables. Let Sn, n[greater-or-equal, slanted]1, be the partial sums of independent random variables with zero means and finite variances and let a(x) be a real function. We present sufficient conditions under which the arithmetic means of a(Snk/(ESnk2)1/2) converge almost surely to for some subsequences {nk, k[greater-or-equal, slanted]1}, where [Phi] denotes the standard normal distribution.

Suggested Citation

  • Rychlik, Zdzislaw & Szuster, Konrad S., 2003. "On strong versions of the central limit theorem," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 347-357, February.
  • Handle: RePEc:eee:stapro:v:61:y:2003:i:4:p:347-357
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(02)00362-0
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Peligrad, Magda & Shao, Qi-Man, 1995. "A note on the almost sure central limit theorem for weakly dependent random variables," Statistics & Probability Letters, Elsevier, vol. 22(2), pages 131-136, February.
    2. Lacey, Michael T. & Philipp, Walter, 1990. "A note on the almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 201-205, March.
    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. Wang, Jiang-Feng & Liang, Han-Ying, 2008. "A note on the almost sure central limit theorem for negatively associated fields," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1964-1970, September.
    2. Miao, Yu & Wang, Rujun & Adler, Andre, 2016. "Limit theorems for order statistics from exponentials," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 51-57.
    3. Xu, Feng & Wu, Qunying, 2017. "Almost sure central limit theorem for self-normalized partial sums of ρ−-mixing sequences," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 17-27.
    4. Denker, Manfred & Zheng, Xiaofei, 2018. "On the local times of stationary processes with conditional local limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2448-2462.
    5. Li, Jingyu & Zhang, Yong, 2021. "An almost sure central limit theorem for the stochastic heat equation," Statistics & Probability Letters, Elsevier, vol. 177(C).
    6. Dudzinski, Marcin, 2003. "A note on the almost sure central limit theorem for some dependent random variables," Statistics & Probability Letters, Elsevier, vol. 61(1), pages 31-40, January.
    7. Chen, Shouquan & Lin, Zhengyan, 2008. "Almost sure functional central limit theorems for weakly dependent sequences," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1683-1693, September.
    8. Wu, Qunying, 2011. "Almost sure limit theorems for stable distributions," Statistics & Probability Letters, Elsevier, vol. 81(6), pages 662-672, June.
    9. Pelletier, Mariane, 1999. "An Almost Sure Central Limit Theorem for Stochastic Approximation Algorithms," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 76-93, October.
    10. Luísa Pereira & Zhongquan Tan, 2017. "Almost Sure Convergence for the Maximum of Nonstationary Random Fields," Journal of Theoretical Probability, Springer, vol. 30(3), pages 996-1013, September.
    11. Ibragimov, Ildar & Lifshits, Mikhail, 1998. "On the convergence of generalized moments in almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 343-351, November.
    12. Bercu, B., 2004. "On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 157-173, May.
    13. Heck, Matthias K., 1998. "The principle of large deviations for the almost everywhere central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 61-75, August.
    14. Holzmann, Hajo & Koch, Susanne & Min, Aleksey, 2004. "Almost sure limit theorems for U-statistics," Statistics & Probability Letters, Elsevier, vol. 69(3), pages 261-269, September.
    15. Horvath, Lajos & Khoshnevisan, Davar, 1995. "Weight functions and pathwise local central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 105-123, September.
    16. István Fazekas & Alexey Chuprunov, 2007. "An Almost Sure Functional Limit Theorem for the Domain of Geometric Partial Attraction of Semistable Laws," Journal of Theoretical Probability, Springer, vol. 20(2), pages 339-353, June.
    17. Matula, Przemyslaw, 2005. "On almost sure limit theorems for positively dependent random variables," Statistics & Probability Letters, Elsevier, vol. 74(1), pages 59-66, August.
    18. Alain Rouault & Marc Yor & Marguerite Zani, 2002. "A Large Deviations Principle Related to the Strong Arc-Sine Law," Journal of Theoretical Probability, Springer, vol. 15(3), pages 793-815, July.
    19. Fahrner, I. & Stadtmüller, U., 1998. "On almost sure max-limit theorems," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 229-236, March.
    20. István Berkes & Siegfried Hörmann & Lajos Horváth, 2010. "On Functional Versions of the Arc-Sine Law," Journal of Theoretical Probability, Springer, vol. 23(1), pages 109-126, March.

    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:61:y:2003:i:4:p:347-357. 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.