Extremes of the standardized Gaussian noise

Abstract

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

18 pages

References in corpus (1)

• Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices