Extremes of the standardized Gaussian noise

Let {ξk,k∈Zd} be a dd-dimensional array of independent standard Gaussian random variables. For a finite set A⊂ZdA⊂Zd define S(A)=∑k∈Aξk. Let |A||A| be the number of elements in AA. We prove that the appropriately normalized maximum of S(A)/|A|, where AA ranges over all discrete cubes or rectangles contained in {1,…,n}d{1,…,n}d, converges in law to the Gumbel extreme-value distribution as n→∞n→∞. We also prove a continuous-time counterpart of this result.