Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different grids

Abstract

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 $ \mathfrak{R}(\delta_1(T))=\{k\delta_1(T): k=0,1,\cdots\}$. In this paper we extend recent findings by considering additionally the \bE{maximum} over another grid $ \mathfrak{R}(\delta_2(T))$. We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting $T\to \infty$.