On Piterbarg’s max-discretisation theorem for multivariate stationary Gaussian processes

Let {X(t),t≥0}$\{X\left(t\right),t\ge 0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004)  23], which we refer to as Piterbarg’s max-discretisation theorem gives the joint asymptotic behaviour (T→∞$T\to \infty$) of the continuous time maximum M(T)=_maxt∈0,T]X(t)$M\left(T\right)={max}_{t\in 0,T]}X\left(t\right)$, and the maximum M^δ(T)=_maxt∈R(δ)X(t)$M^{\delta }\left(T\right)={max}_{t\in \mathfrak{R}\left(\delta \right)}X\left(t\right)$, with R(δ)⊂0,T]$\mathfrak{R}\left(\delta \right)\subset 0,T]$ a uniform grid of points of distance δ=δ(T)$\delta =\delta \left(T\right)$. 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 δ(T)$\delta \left(T\right)$ approaches 0 as T→∞$T\to \infty$. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.