  Let $\{\xi_n, n\in\Z^d\}$ be a $d$-dimensional array of i.i.d. Gaussian
random variables and define $\SSS(A)=\sum_{n\in A} \xi_n$, where $A$ is a
finite subset of $\Z^d$. We prove that the appropriately normalized maximum of
$\SSS(A)/\sqrt{|A|}$, where $A$ ranges over all discrete cubes or rectangles
contained in $\{1,\ldots,n\}^d$, converges in the weak sense to the Gumbel
extreme-value distribution as $n\to\infty$. We also prove continuous-time
counterparts of these results.
