Some Characteristic Properties of Stochastic Gaussian Processes

A. M. Veršik

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 390-394

Résumé

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.