Teoriya Veroyatnostei i ee Primeneniya
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Guidelines for authors
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Teor. Veroyatnost. i Primenen.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Teoriya Veroyatnostei i ee Primeneniya, 1966, Volume 11, Issue 3, Pages 514–518 (Mi tvp648)  

This article is cited in 11 scientific papers (total in 11 papers)

Short Communications

On a relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm

V. V. Petrov

Leningrad
Abstract: Let $\{X_n\}$ $(n=1,2,\dots)$ be a sequence of independent random variables having zero means and finite variances. Let us denote
\begin{gather*} S_n=\sum_{j=1}^nX_j,\quad B_n=\sum_{j=1}^n\mathbf E(X_j^2), \\ R_n=\sup_{-\infty<x<\infty}\biggl|\mathbf P(S_n<x\sqrt{B_n})-\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-t^2/2}\,dt\biggr|. \end{gather*}
The following theorem is proved.
Theorem 1. {\it Suppose that
\begin{gather*} B_n\to\infty,\quad\frac{B_{n+1}}{B_n}\to1, \\ R_n=O\biggl(\frac1{(\ln B_n)^{1+\delta}}\biggr)\quad\text{for some }\delta>0. \end{gather*}
Then
$$ \mathbf P\biggl(\limsup\frac{S_n}{(2B_n\ln\ln B_n)^{1/2}}=1\biggr)=1. $$
}
Received: 19.10.1965
English version:
Theory of Probability and its Applications, 1966, Volume 11, Issue 3, Pages 454–458
DOI: https://doi.org/10.1137/1111046
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: V. V. Petrov, “On a relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm”, Teor. Veroyatnost. i Primenen., 11:3 (1966), 514–518; Theory Probab. Appl., 11:3 (1966), 454–458
Citation in format AMSBIB
\Bibitem{Pet66}
\by V.~V.~Petrov
\paper On a~relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm
\jour Teor. Veroyatnost. i Primenen.
\yr 1966
\vol 11
\issue 3
\pages 514--518
\mathnet{http://mi.mathnet.ru/tvp648}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=212855}
\zmath{https://zbmath.org/?q=an:0203.19602}
\transl
\jour Theory Probab. Appl.
\yr 1966
\vol 11
\issue 3
\pages 454--458
\crossref{https://doi.org/10.1137/1111046}
Linking options:
  • https://www.mathnet.ru/eng/tvp648
  • https://www.mathnet.ru/eng/tvp/v11/i3/p514
  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024