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An almost everywhere central limit theorem. (English) Zbl 0668.60029
The main purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem. The main result gives the following Theorem: Let \(Y_ n\), \(n\geq 1\), be i.i.d. random variables on (\(\Omega\),U,P) with E \(Y_ 1=0\), E \(Y^ 2_ 1=1\) and E \(| Y_ 1|^{2+2\delta}<\infty\) for some \(\delta >0\). Let \(S_ n=\sum^{n}_{k=1}Y_ k\), \(n\geq 1\). Then P-a.e. \[ \lim_{n\to \infty}(\log n)^{-1}\sum^{n}_{k=1}k^{-1}\delta_{S_ k(w)/\sqrt{k}}=N(0,1), \] where N(0,1) is the standard normal distribution on R, and the convergence is weak convergence of measures on R.
Reviewer: Z.Rychlik

60F05 Central limit and other weak theorems
60F15 Strong limit theorems
Full Text: DOI
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