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

Let { X ( t ) , t ≥ 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 → ∞ ) 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.