Upper Bounds for the Asymptotic Maxima of Continuous Gaussian Processes

Upper bounds are obtained for $|X(t)|/Q(t)$ as $t ightarrow infty$, where $X(t)$ is a continuous Gaussian process with $EX^2(t) leqq Q^2(t), Q(t)$ non-decreasing. Our results are extensions of some work of Pickands (1967), Nisio (1967) and Orey (1971) to larger classes of Gaussian processes, i.e. fewer restrictions are imposed on the covariance functions. The results follow from Fernique's lemma (1964) and a recent lemma on the maximum of Gaussian sequences due to Landau, Shepp, Fernique and the author (see Marcus, Shepp (1971) for further references to this lemma).