Limit theorems for excursion sets of stationary random fields

TL;DR

This paper reviews recent limit theorems concerning the geometric properties of excursion sets in stationary random fields, including Gaussian and stable types, with implications for statistical testing and asymptotic analysis.

Contribution

It provides a comprehensive overview of new asymptotic results and limit theorems for the geometry of excursion sets across various classes of stationary random fields.

Findings

Limit theorems for volume of excursion sets of various random fields.

Functional limit theorems involving Gaussian processes.

Results on surface area of excursion sets for Gaussian fields.

Abstract

We give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we cover a number of limit theorems of central type for the volume of excursions of stationary (quasi--, positively or negatively) associated random fields with stochastically continuous realizations for a fixed excursion level. This class includes in particular Gaussian, Poisson shot noise, certain infinitely divisible, $\alpha$--stable and max--stable random fields satisfying some extra dependence conditions. Functional limit theorems (with the excursion level being an argument of the limiting Gaussian process) are reviewed as well. For stationary isotropic $C^1$--smooth Gaussian random fields similar results are available also for the surface area of the excursion set. Statistical tests of Gaussianity of a random field which are of importance to real data analysis as well as results for an increasing excursion level round up the paper.