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

The subject of this work is the multivariate generalization of the theory of multiple Wiener–Itô integrals. In the scalar valued case this theory was described by the author in 2014. The proofs of the present paper apply the technique of that work, but in the proof of some results new ideas were needed. The motivation for this study was a result in the paper “Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors” (1994) by Arcones, which contained the multivariate generalization of a non-central limit theorem for non-linear functionals of Gaussian stationary random fields presented in a paper by R. L. Dobrushin and the author. However, the formulation of Arcones' result was incorrect. To present it in a correct form, the multivariate version of the theory explained in my work of 2014 has to be worked out, because the notions introduced in this theory are needed in its formulation. This is done in the present paper. In its continuation it will be explained how to work out a method with the help of the results in this work that enables us to prove non-Gaussian limit theorems for non-linear functionals of vector valued Gaussian stationary random fields. The right version of Arcones' result presented also in the introduction of this work will be formulated and proved with its help in a future paper of mine.