  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%)
