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 -> infinity) of the continuous time maximum M(T) = max(t is an element of[0,T]) X(t), and the maximum M-delta(T) = max(t is an element of R(delta)) X(t), with R(delta) subset of [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 -> infinity. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes