Some Characteristic Properties of Stochastic Gaussian Processes

In the paper spherically invariant processes are defined. The characteristic function of these processes $(\xi(t))$ in accordance with Shonberg's theorem [1] is of the form
$$ \chi(\eta)\equiv{\mathbf M}e^{i\eta}=f({\mathbf D}\eta)=\int_0^\infty e^{-\gamma{\mathbf D}\eta} G(d\gamma), $$
$\eta=\int\xi(t)\eta(t) dt$, where $G$ is some measure on $[0,\infty)$. Only if the process is spherically invariant, then 1) every extrapolation problem has a linear solution, 2) every functional transformation leaving the correlation function of the process invariant retains its measure in the space of realizations.If a spherically invariant process is stationary and ergodic, then it is Gaussian.