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Considering a sequence of standardized stationary Gaussian random variables, a universal result in the almost sure central limit theorem for maxima and partial sum is established. Our result generalizes and improves that on the almost sure central limit theory previously obtained by Marcin Dudzinski [1]. Our result reaches the optimal form.

A Note on the Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences