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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, 11:3, 454–458

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