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Teor. Veroyatnost. i Primenen., 1964, Volume 9, Issue 2, Pages 390–394 (Mi tvp388)  

This article is cited in 5 scientific papers (total in 6 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.

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English version:
Theory of Probability and its Applications, 1964, 9:2, 353–356

Bibliographic databases:

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

Citation in format AMSBIB
\Bibitem{Ver64}
\by A.~M.~Ver{\v s}ik
\paper Some Characteristic Properties of Stochastic Gaussian Processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1964
\vol 9
\issue 2
\pages 390--394
\mathnet{http://mi.mathnet.ru/tvp388}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=165577}
\zmath{https://zbmath.org/?q=an:0141.15203}
\transl
\jour Theory Probab. Appl.
\yr 1964
\vol 9
\issue 2
\pages 353--356
\crossref{https://doi.org/10.1137/1109053}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Arnol'd, M. Sh. Birman, I. M. Gel'fand, I. A. Ibragimov, S. V. Kerov, A. A. Kirillov, O. A. Ladyzhenskaya, G. A. Leonov, A. A. Lodkin, S. P. Novikov, Ya. G. Sinai, M. Z. Solomyak, L. D. Faddeev, “Anatolii Moiseevich Vershik (on his sixtieth birthday)”, Russian Math. Surveys, 49:3 (1994), 207–221  mathnet  crossref  mathscinet  adsnasa  isi
    2. Theory Probab. Appl., 61:3 (2017), 375–388  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Theory Probab. Appl., 62:2 (2018), 335–338  mathnet  crossref  crossref  mathscinet  isi  elib
    4. Ll'inskii A., “On Notions of Q-Independence and Q-Identical Distributiveness”, Stat. Probab. Lett., 140 (2018), 33–36  crossref  isi
    5. Theory Probab. Appl., 63:3 (2019), 393–407  mathnet  crossref  crossref  isi  elib
    6. Zaitsev A.Yu. Kagan A.M. Nikitin Ya.Yu., “Toward the History of the St. Petersburg School of Probability and Statistics. Iv. Characterization of Distributions and Limit Theorems in Statistics”, Vestn. St Petersb. Univ.-Math., 52:1 (2019), 36–53  crossref  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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