On Piterbarg Max-discretisation Theorem for Multivariate Stationary Gaussian Processes

Abstract

Let $\{X(t), t\geq0\}$ 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.