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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1962, Volume 7, Issue 2, Pages 204–208 (Mi tvp4715)

Short Communications

On the Structure of the Infinitesimal $\sigma$-Algebra of a Gaussian Process

V. N. Tutubalin, M. I. Freidlin

Moscow

Abstract: Let $x(t)$ be a Gaussian stationary process $\mathfrak{M}_{+0}=\bigcap _{t>0}\mathfrak{M}_t$, where $\mathfrak{M}_t$ is the $\sigma$-algebra generated by $x(s),0\leq s\leq t$. It is proved that if the spectral density $f(\lambda)$ of the process satisfies the condition $f(\lambda)\geq{1}/{\lambda^p}$ for all $|\lambda|>\lambda_0$ and some $p>0$, the $\sigma$-algebra $\mathfrak{M}_{+0}$ is generated by $x(0),{dx(0)}/{dt},…,{dx^{(k)}{(0)}}/{dt^k}$, where $k$ is the order of the derivative the sample functions admit.

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English version:
Theory of Probability and its Applications, 1962, 7:2, 196–199

Citation: V. N. Tutubalin, M. I. Freidlin, “On the Structure of the Infinitesimal $\sigma$-Algebra of a Gaussian Process”, Teor. Veroyatnost. i Primenen., 7:2 (1962), 204–208; Theory Probab. Appl., 7:2 (1962), 196–199

Citation in format AMSBIB
\Bibitem{TutFre62} \by V.~N.~Tutubalin, M.~I.~Freidlin \paper On the Structure of the Infinitesimal $\sigma$-Algebra of a Gaussian Process \jour Teor. Veroyatnost. i Primenen. \yr 1962 \vol 7 \issue 2 \pages 204--208 \mathnet{http://mi.mathnet.ru/tvp4715} \transl \jour Theory Probab. Appl. \yr 1962 \vol 7 \issue 2 \pages 196--199 \crossref{https://doi.org/10.1137/1107019}