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A note on the almost sure central limit theorem. (English) Zbl 0691.60016
For broad classes of sequences $$\{X_ j$$, $$j\geq 1\}$$ of weakly dependent r.v.’s almost sure invariance principles (ASIP) are known e.g. in the form $\sum_{j\leq n}X_ j-\sum_{j\leq n}Y_ j=o(n^{1/2})\text{ with probability 1 }(n\to \infty),$ where $$\{Y_ j$$, $$j\geq 1\}$$ is a sequence of i.i.d. N(0,1) r.v.’s. The behaviour of the k-th partial sums $$S_ k$$ of $$\{X_ j$$, $$j\geq 1\}$$ is studied by the help of a summation method (logarithmic mean), if an ASIP is supposed. A new proof of an almost sure central limit theorem (without ergodic theory), and some extensions and further remarks on known special results in this field are given.