Some examples of equivalent and orthogonal Gaussian distributions

Let $t = (t_1,\dots, t_N)\subset T\in E^N$, $\lambda=(\lambda_1,\dots,\lambda_N)\in E^N$, $E^N$ be the $N$-dimensional Euclidean space, $\xi_i\colon T \to E^1$, $i=1,2$, be homogeneous Gaussian fields, and let $\nu_1$, $\nu_2$ be measures induced by $\xi_1$, $\xi_2$. The random fields $\xi_1$ and $\xi_2$ are supposed to have the rational spectral densities. Three examples illustrating the difference between the cases $N=1$ and $N\ge 2$ are given.