IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v8y1978i2p181-201.html
   My bibliography  Save this article

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Author

Listed:
  • Butzer, P. L.
  • Hahn, L.

Abstract

Let (Xn)n[epsilon] be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions FXn, and Sn = [Sigma]i=1n Xi. The authors present limit theorems together with convergence rates for the normalized sums [phi](n)Sn, where [phi]: --> +, [phi](n) --> 0, n --> [infinity], towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression [is proportional to]f(x) d[F[phi](n)Sn(x) - FX(x)] which depend upon the normalizing function [phi], decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.

Suggested Citation

  • Butzer, P. L. & Hahn, L., 1978. "General theorems on rates of convergence in distribution of random variables I. General limit theorems," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 181-201, June.
    • Handle: RePEc:eee:jmvana:v:8:y:1978:i:2:p:181-201
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0047-259X(78)90071-4
    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. Yu Zhang, 2023. "Asymptotic Normality of M-Estimator in Linear Regression Model with Asymptotically Almost Negatively Associated Errors," Mathematics, MDPI, vol. 11(18), pages 1-16, September.

    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:jmvana:v:8:y:1978:i:2:p:181-201. 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.