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Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 3, Pages 514–518 (Mi tvp648)  

This article is cited in 4 scientific papers (total in 4 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 $\{X_n\}$ $(n=1,2,…)$ 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}|\mathbf P(S_n<x\sqrt{B_n})-\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-t^2/2} dt|. \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(\frac1{(\ln B_n)^{1+\delta}})\quadfor some \delta>0. \end{gather*}
Then
$$ \mathbf P(\limsup\frac{S_n}{(2B_n\ln\ln B_n)^{1/2}}=1)=1. $$
}

Full text: PDF file (273 kB)

English version:
Theory of Probability and its Applications, 1966, 11:3, 454–458

Bibliographic databases:

Received: 19.10.1965

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://www.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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Bourgade P., Hughes C.P., Nikeghbali A., Yor M., “The characteristic polynomial of a random unitary matrix: A probabilistic approach”, Duke Mathematical Journal, 145:1 (2008), 45–69  crossref  mathscinet  isi
    2. Fukuyama K., Ueno Y., “On the central limit theorem and the law of the iterated logarithm”, Statistics & Probability Letters, 78:12 (2008), 1384–1387  crossref  mathscinet  zmath  isi
    3. Han G., “Limit Theorems in Hidden Markov Models”, IEEE Trans. Inf. Theory, 59:3 (2013), 1311–1328  crossref  isi
    4. M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163  crossref  crossref  mathscinet  zmath  isi  elib  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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