New result on the behaviour of Gaussian maxima in terms of the covariance function

It is a well-known result by Berman [1] 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 $\sqrt{2\log(n)}(1+o(1))$ almost surely. In this work we discuss whether or not the Cesàro convergence of $|r(n)|$ to zero necessarily implies the same asymptotic.