The theory of Wiener–Itô integrals in vector-valued Gaussian stationary random fields. Part II

This work is the continuation of my paper in Moscow Math. Journal Vol. 20, No. 4 in 2020. In that paper the existence of the spectral measure of a vector-valued stationary Gaussian random field is proved and the vector-valued random spectral measure corresponding to this spectral measure is constructed. The most important properties of this random spectral measure are formulated, and they enable us to define multiple Wiener–Itô integrals with respect to it. Then an important identity about the products of multiple Wiener–Itô integrals, called the diagram formula is proved. In this paper an important consequence of this result, the multivariate version of Itô's formula is presented. It shows a relation between multiple Wiener–Itô integrals with respect to vector-valued random spectral measures and Wick polynomials. Wick polynomials are the multivariate versions of Hermite polynomials. With the help of Itô's formula the shift transforms of a random variable given in the form of a multiple Wiener–Itô integral can be written in a useful form. This representation of the shift transforms makes possible to rewrite certain non-linear functionals of a vector-valued stationary Gaussian random field in such a form which suggests a limiting procedure that leads to new limit theorems. Finally, this paper contains a result about the problem when this limiting procedure may be carried out, i.e., when the limit theorems suggested by our representation of the investigated non-linear functionals are valid.