Extremes of the standardized Gaussian noise

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