THE THEORY OF WIENER-IT Ó INTEGRALS IN VECTOR-VALUED GAUSSIAN STATIONARY RANDOM FIELDS

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-Ito integrals with respect to it. Then an important identity about the products of multiple Wiener-Ito integrals, called the diagram formula is proved. In this paper an important consequence of this result, the multivariate version of Ito's formula is presented. It shows a relation between multiple Wiener-Ito 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 Ito's formula the shift transforms of a random variable given in the form of a multiple Wiener-Ito 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. 2020 MATH. SUBJ. CLASS. 60G10, 60G15, 60F99.