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Let {X(t), t ≥ 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 → ∞) of the continuous time maximum M (T ) = max t∈[0,T ] X(t), and the maximum M δ (T ) = max t∈R(δ) X(t),with R(δ) ⊂ [0, T ] a uniform grid of points of distance δ = δ(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 δ(T ) approaches 0 as T → ∞. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.

In this paper we derive Piterbarg's max-discretisation theorem for two different grids considering centered stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over [0, T ] and over a grid R(δ 1 (T )) = {kδ 1 (T ) : k = 0, 1, · · · }. In this paper we extend the recent findings by considering additionally the maximum over another grid R(δ 2 (T )). We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting T → ∞. As a by-product we find that the joint limiting distribution of the maximum over different grids, which we refer to as the Piterbarg distribution, is in the case of weakly dependent Gaussian processes a max-stable distribution.