New Result on the Behavior of Gaussian Maxima in Terms of the Covariance Function

It is a well-known result by Berman (1964) that if the covariance function r(n) of a stationary centered Gaussian sequence tends to zero as n tends to infinity, then the maximum of its first n elements is \(\sqrt{2 \text{log}\left(n\right)}\left(1+o\left(1\right)\right)\) $\sqrt{2\text{log}\left(n\right)}\left(1,+,o,\left(1\right)\right)$ almost surely. In this paper, we discuss whether or not the Cesàro convergence of |r(n)| to zero necessarily implies the same asymptotic.