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Teor. Veroyatnost. i Primenen., 1964, Volume 9, Issue 2, Pages 386–390 (Mi tvp387)  

Short Communications

On the Asymptotic Behaviour of the Estimate of the Spectral Function for a Stationary Gaussian Process

T. L. Malevič

Taškent

Abstract: Let $\xi _n$, $n=0$, $\pm 1,\pm 2,…$, be a real stationary Gaussian sequence with an absolutely continuous spectral function $F(\lambda)$, and let $F_N (\lambda)$ be the sample spectral function.We assume that $F(\lambda)$ has no interval of constancy, and $f(\lambda)=F'(\lambda)\in L_2[0,\pi]$. Then the sequence of measures $P_N$ generated by the process $\zeta_N(\lambda)=\sqrt N[F_n(\lambda)-F(\lambda)]$ converges weakly to the measure which is generated by the Gaussian process $\zeta(\lambda)$ with ${\mathbf M}\zeta(\lambda)=0$ and
$$ {\mathbf M}\zeta(\lambda)\zeta(\mu)=2\pi\int_0^{\min(\lambda\mu)}f^2(x) dx. $$
A similar result holds for the process $\xi_t$ with continuous time, $0\leqslant t<+\infty$.

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English version:
Theory of Probability and its Applications, 1964, 9:2, 350–353

Bibliographic databases:

Received: 22.06.1963

Citation: T. L. Malevič, “On the Asymptotic Behaviour of the Estimate of the Spectral Function for a Stationary Gaussian Process”, Teor. Veroyatnost. i Primenen., 9:2 (1964), 386–390; Theory Probab. Appl., 9:2 (1964), 350–353

Citation in format AMSBIB
\Bibitem{Mal64}
\by T.~L.~Malevi{\v{c}}
\paper On the Asymptotic Behaviour of the Estimate of the Spectral Function for a~Stationary Gaussian Process
\jour Teor. Veroyatnost. i Primenen.
\yr 1964
\vol 9
\issue 2
\pages 386--390
\mathnet{http://mi.mathnet.ru/tvp387}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=163350}
\zmath{https://zbmath.org/?q=an:0132.38401}
\transl
\jour Theory Probab. Appl.
\yr 1964
\vol 9
\issue 2
\pages 350--353
\crossref{https://doi.org/10.1137/1109052}


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