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

TL;DR

This paper extends Piterbarg's max-discretisation theorem to stationary vector Gaussian processes observed on two different grids, deriving the joint limiting distribution of their maxima as observation window grows large.

Contribution

It introduces a new theoretical result for the joint distribution of maxima over two different observation grids for vector Gaussian processes, expanding previous single-grid analyses.

Findings

Derived joint limiting distribution for maxima over two grids

Extended Piterbarg's theorem to vector Gaussian processes

Applicable for large observation windows T

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 $T\to \infty$.