Extremes of a type of locally stationary Gaussian random fields with applications to Shepp statistics

Abstract

Let $\{Z(\tau ,s),(\tau ,s)\in [a,b]\times [0,T]\}$ with some positive constants $a,b,T$ be a centered Gaussian random field with variance function ${\sigma }^2(\tau ,s)$ satisfying ${\sigma }^2(\tau ,s)={\sigma }^2(\tau )$. We firstly derive the exact tail asymptotics for the maximum $M_H(T)={\max }_{(\tau ,s)\in [a,b]\times [0,T]}Z(\tau ,s)/\sigma (\tau )$ up crossing some level $u$ with any fixed $0<a<b<\infty$ and $T>0$; and we further derive the extreme limit law for $M_H(T)$. As applications of the main results, we derive the exact tail asymptotics and the extreme limit law for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as input.

15 pages,0 figures

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