# Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

## TL;DR

This paper investigates the joint limiting distributions of maxima and minima of centered homogeneous Gaussian random fields over continuous time and various grids, revealing dependence structures based on the field's dependence strength.

## Contribution

It provides new theoretical results on the asymptotic dependence or independence of maxima and minima for different types of Gaussian fields and grid configurations.

## Key findings

- Maxima and minima are asymptotically dependent for strongly dependent fields with sparse, Pickands', or dense grids.
- Maxima and minima are asymptotically independent for weakly dependent fields.
- The results extend understanding of extremal behavior in Gaussian random fields.

## Abstract

In this paper, for centered homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.