On Piterbarg Max-discretisation Theorem for Multivariate Stationary Gaussian Processes

Abstract

Let $\{X(t),t\ge 0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004), which we refer to as Piterbarg's max-discretisation theorem gives the joint asymptotic behaviour ($T\to \infty$) of the continuous time maximum $M(T)={\max }_{t\in [0,T]}X(t),$ and the maximum $M^{\delta }(T)={\max }_{t\in \mathfrak{R}(\delta )}X(t),$ with $\mathfrak{R}(\delta )\subset [0,T]$ a uniform grid of points of distance $\delta =\delta (T)$. Under some asymptotic restrictions on the correlation function Piterbarg's max-discretisation theorem shows that for the limit result it is important to know the speed $\delta (T)$ approaches 0 as $T\to \infty$. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.

References in corpus (2)

• Limit Laws for Extremes of Dependent Stationary Gaussian Arrays
• Asymptotics of maxima of strongly dependent Gaussian processes

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• Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different grids
• On maxima of chi-processes over threshold dependent grids
• Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids