IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v27y1996i3p247-254.html
   My bibliography  Save this article

Laws of the iterated logarithm for weighted sums of independent random variables

Author

Listed:
  • Li, Deli
  • Tomkins, R. J.

Abstract

Let [Lambda] = lim supn-->[infinity](2n log log n)-1/2 [Sigma]k=1n[latin small letter f with hook](k/n)Xk, where [latin small letter f with hook] is a function defined on [0,1] and {X, Xn;n[greater-or-equal, slanted]1} is an iid sequence. If X is real-valued, it is shown that [Lambda] = [latin small letter f with hook]2, the L2-norm of [latin small letter f with hook], for all functions [latin small letter f with hook] in a certain class of absolutely continuous functions if E(X) = 0 and E(X2) = 1. Conversely, if [Lambda] = [latin small letter f with hook]2 for some such [latin small letter f with hook] with [integral operator]01[latin small letter f with hook](t)dt [not equal to] 0, then E(X) = 0, E(X2) = 1. Necessary and sufficient conditions for the compact law of the iterated logarithm are given in the case when X takes values in a separable Banach space, and a law of the iterated logarithm for sums of weighted partial sums is obtained in a Banach space setting.

Suggested Citation

  • Li, Deli & Tomkins, R. J., 1996. "Laws of the iterated logarithm for weighted sums of independent random variables," Statistics & Probability Letters, Elsevier, vol. 27(3), pages 247-254, April.
  • Handle: RePEc:eee:stapro:v:27:y:1996:i:3:p:247-254
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0167-7152(95)00072-0
    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.

    Citations

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


    Cited by:

    1. Li, Deli & Qi, Yongcheng & Rosalsky, Andrew, 2009. "Iterated logarithm type behavior for weighted sums of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 643-651, March.
    2. Pingyan, Chen, 2002. "Limiting behavior of weighted sums with stable distributions," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 367-375, December.
    3. Peng, Liang & Qi, Yongcheng, 2003. "Chover-type laws of the iterated logarithm for weighted sums," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 401-410, December.
    4. Li, Deli & Bhaskara Rao, M. & Tomkins, R. J., 2001. "The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 78(2), pages 191-217, August.
    5. Zuoxiang Peng & Zhongquan Tan & Saralees Nadarajah, 2011. "Almost sure central limit theorem for the products of U-statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 73(1), pages 61-76, January.

    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:27:y:1996:i:3:p:247-254. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.