A note on the functional law of iterated logarithm for maxima of Gaussian sequences

Let {Xn, n >= 1} be a real-valued stationary Gaussian sequence with mean zero and variance one. Let Mn = max{Xt, i <= n} and Hn(t) = (M[nt] - bn)an-1 be the maximum resp. the properly normalised maximum process, where cn = (2 log n)1/2, an = (log log n)/cn and . We characterize the almost sure limit functions of (Hn)n>=3 in the set of non-negative, non-decreasing, right-continuous, real-valued functions on (0, [infinity]), if r(n) (log n)3-[Delta] = O(1) for all [Delta] > 0 or if r(n) (log n)2-[Delta] = O(1) for all [Delta] > 0 and r(n) convex and fulfills another regularity condition, where r(n) is the correlation function of the Gaussian sequence.