On Piterbarg's max-discretisation theorem for homogeneous Gaussian random fields

TL;DR

This paper investigates the asymptotic relationship between the maximum of a continuous homogeneous Gaussian random field and its sampled maxima at discrete points, revealing dependence patterns based on grid density and dependence strength.

Contribution

It extends Piterbarg's max-discretisation theorem to homogeneous Gaussian fields, characterizing the asymptotic dependence of maxima under various grid densities and dependence conditions.

Findings

Weakly dependent case: maxima are asymptotically independent, dependent, or coincide depending on grid density.

Strongly dependent case: maxima are asymptotically totally dependent on dense grids.

Results unify and extend previous theorems for Gaussian processes to fields.

Abstract

Motivated by the papers of Piterbarg (2004) and H\"{u}sler (2004), in this paper the asymptotic relation between the maximum of a continuous dependent homogeneous Gaussian random field and the maximum of this field sampled at discrete time points is studied. It is shown that, for the weakly dependent case, these two maxima are asymptotically independent, dependent and coincide when the grid of the discrete time points is a sparse grid, Pickands grid and dense grid, respectively, while for the strongly dependent case, these two maxima are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense, and asymptotically dependent if the the grid points are sparse or Pickands grids.