On Piterbarg Max-discretisation Theorem for Multivariate Stationary Gaussian Processes

TL;DR

This paper extends Piterbarg's max-discretisation theorem to multivariate stationary Gaussian processes, analyzing the joint asymptotic behavior of continuous and discretized maxima as the observation window grows.

Contribution

It generalizes Piterbarg's theorem from univariate to multivariate Gaussian processes, providing new asymptotic results under specific correlation conditions.

Findings

Joint asymptotic behavior of maxima derived for multivariate processes

Conditions on grid spacing $oldsymbol{ o 0}$ as $T o oldsymbol{ o ext{infinity}}$

Extension of Piterbarg's theorem to multivariate setting

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.