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On strong versions of the central limit theorem. (English) Zbl 0661.60031
Let $$X_ 1,X_ 2,...$$, be i.i.d. r.v.’s with E $$X_ 1=0$$, E $$X^ 2_ 1=1$$, and let $$S_ n=X_ 1+...+X_ n$$, $$n\geq 1$$. The main results of the paper are as follows: $i)\quad P[\lim_{N}\sup_{x}| N^{- 1}\sum^{N}_{n=1}\mathbf{1}_{(-\infty,x)}(S_ n/\sqrt{n})-\Phi (x)| =0]=0,$ where $$\Phi$$ is the standard normal distribution function.
ii) Let a(.) be a bounded and continuous function at each point except a finite number of points on the real line. If E $$| X_ 1|^ 3<\infty$$, then $P[\lim_{N}(\log N)^{-1}/\sum^{N}_{n=1}n^{- 1}a(S_ n/\sqrt{n})=\int_{R}a(y)d\Phi (y)]=1.$ In particular, for $$a=\mathbf{1}_{(-\infty,x)}$$ the limit above equals $$\Phi$$ (x) and the convergence in ii) is uniform w.r.t. $$x\in R$$. A similar result for arithmetic means of appropriate subsequences is also obtained.
Remark: It seems that for the proof of ii) ess sup of a(.) on each subinterval of a finite partition of R should be used. The presented example shows that ii) does not hold if a(.) is discontinuous at infinitely many points.
Reviewer: A.M.Zapala

##### MSC:
 60F05 Central limit and other weak theorems 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks
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##### References:
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