Limit theorems for non-linear functionals of stationary Gaussian random fields

TL;DR

This paper discusses limit theorems for non-linear functionals of stationary Gaussian random fields, providing a detailed theoretical framework, spectral analysis, and applications including non-Gaussian limit results.

Contribution

It introduces a comprehensive spectral representation approach and constructs multiple Wiener--Itô integrals for analyzing non-linear functionals of Gaussian fields.

Findings

Spectral measure representation of covariance functions

Construction and properties of multiple Wiener--Itô integrals

Non-Gaussian limit theorems for functionals of Gaussian fields

Abstract

This is an extended version of a series of talks I held at the University of Bochum in 2017 about limit theorems for non-linear functionals of stationary Gaussian random fields. The goal of these talks was to give a fairly detailed introduction to the theory leading to such results, even if some of the results are presented without proof. On the other hand, I gave a simpler proof for some of the results. (The proofs omitted from this text can be found in my Springer Lecture Note Multiple Wiener--Ito Integrals. In this note first I discuss the spectral representation of the covariance function of a Gaussian stationary rendom field by means of the spectral measure and the representation of the elements of the random field by means of a random integral with respect to the random spectral measure. Then I construct the multiple random integrals with respect to the random spectral measure and prove their most important properties. Finally I show some interesting applications of these multiple random integrals. In particular, I prove some non-trivial non-Gaussian limit theorems