# Wiener--Ito integral representation in vector valued Gaussian stationary random fields

## TL;DR

This paper extends the theory of Wiener--Itô integrals to multivariate Gaussian stationary random fields, providing foundational results necessary for analyzing non-linear functionals and completing proofs of related limit theorems.

## Contribution

It offers a multivariate generalization of Wiener--Itô integrals, filling gaps in previous proofs and establishing foundational properties for vector-valued Gaussian fields.

## Key findings

- Developed a multivariate Wiener--Itô integral framework
- Provided detailed properties of non-linear functionals of vector Gaussian fields
- Completed a full proof of a multivariate non-central limit theorem

## Abstract

The subject of this work is the multivariate generalization of the theory of multiple Wiener--It\^o integrals. In the scalar valued case this theory was described in paper\cite{11}. Our proofs apply the technique of this work, but in the proof of some results new ideas were needed. The motivation for this study was a result in paper\cite{1} of Arcones where he formulated the multivariate version of a non-central limit theorem for non-linear functionals of Gaussian stationary random fields presented in paper\cite{6}. We found the proof in paper\cite{1} incomplete and wanted to give a full proof. We did it in paper\cite{13}, but in that proof we needed a detailed description of the properties of non-linear functionals of vector valued stationary Gaussian fields. Here we provide the foundation needed to carry out that proof. --More--(0%)