Extremes of the standardized Gaussian noise

TL;DR

This paper investigates the asymptotic behavior of the maximum normalized sum of Gaussian noise over discrete cubes and rectangles, demonstrating convergence to the Gumbel distribution, with extensions to continuous-time cases.

Contribution

It establishes the weak convergence of the maximum of normalized Gaussian sums over multidimensional discrete and continuous regions to the Gumbel distribution, extending extreme value theory.

Findings

Maximum normalized sums converge to Gumbel distribution

Results hold for multidimensional discrete cubes and rectangles

Continuous-time analogues are also proven

Abstract

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.