A Note for Extension of Almost Sure Central Limit Theory

for any continuity point x of H . Several papers have dealt with logarithmic limit theorems of this kind and the above relation has been extended in various directions. Fahrner and Stadtmüller [5] gave an almost sure version of a maximum limit theorem. Berkes and Horváth [2] obtained a strong approximation for the logarithmic average of sample extremes. Berkes and Csáki [1] showed that not only the central limit theorem, but every weak limit theorem for independent random variables has an analogous almost sure version. For stationary Gaussian sequences with covariance rn, Csáki and Gonchigdanzan [3] proved an almost sure limit theorem for the maxima of the sequences under the condition rn log n(log logn) 1+ε = O(1). For some dependent random variables, Peligrad and Shao [7] and Dudziński [4] obtained corresponding results about the almost sure central limit theorem.