Mittal and Ylvisaker have shown a law of iterated logarithms for the maxima of certain stationary Gaussian processes. Their result is extended in this paper in that it is shown to be valid under weaker mixing conditions.
"An Extension to a Strong Law Result of Mittal and Ylvisaker for the Maxima of Stationary Gaussian Processes." Ann. Probab. 8 (3) 498 - 510, June, 1980. https://doi.org/10.1214/aop/1176994724