Extremes of the standardized Gaussian noise

Let be a d-dimensional array of independent standard Gaussian random variables. For a finite set define . Let A be the number of elements in A. We prove that the appropriately normalized maximum of , 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-->[infinity]. We also prove a continuous-time counterpart of this result.

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