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Teoriya Veroyatnostei i ee Primeneniya, 1964, Volume 9, Issue 2, Pages 390–394
(Mi tvp388)
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This article is cited in 79 scientific papers (total in 80 papers)
Short Communications
Some Characteristic Properties of Stochastic Gaussian Processes
A. M. Veršik Leningrad
Abstract:
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.
Received: 21.10.1963
Citation:
A. M. Veršik, “Some Characteristic Properties of Stochastic Gaussian Processes”, Teor. Veroyatnost. i Primenen., 9:2 (1964), 390–394; Theory Probab. Appl., 9:2 (1964), 353–356
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