A note on the almost sure central limit theorem (Q582676)

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.