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

V. V. Petrov

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 3, pp. 514-518

Résumé

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 $$ \mathbf P\biggl(\limsup\frac{S_n}{(2B_n\ln\ln B_n)^{1/2}}=1\biggr)=1. $$}