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

Abstract

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)=maxt∈[0,T]X(t), and the maximum Mδ(T)=maxt∈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.