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Teor. Veroyatnost. i Primenen., 1965, Volume 10, Issue 3, Pages 500–509 (Mi tvp545)  

Short Communications

Some properties of estimators of the spectrum of a stationary process

T. L. Malevich

Tashkent

Abstract: Let $x_n$ ($n=0,\pm1,\pm2,…$) be a real Gaussian stationary process with $\mathbf Ex_n=0$ and with the spectral function $F(\lambda)$ which is unknown and is supposed to be continuous.
The statistic
$$ F_N(\lambda)=\frac1{2\pi N}\int_0^\lambda|\sum_{n=1}^Nx_ne^{-iny}|^2 dy $$
is used as an estimator of $F(\lambda)$.
In § 1 estimations of the moments $\mathbf E\max\limits_{0\le\lambda\le\pi}|F_N(\lambda)-F(\lambda)|^k$ are obtained. For example the following theorem holds true.
Theorem 1.3. For the process $x_n$
$$ \mathbf E\max_{0\le\lambda\le\pi}|F_N(\lambda)-F(\lambda)|^k\le C^kk![\omega_F(\frac1N)]^{\frac k2}, $$
where $\omega_F(\cdot)$ is the modulus of continuity of $F(\lambda)$.
In § 2 the probability of large deviations of $F_N(\lambda)$ from $F(\lambda)$ is studied.
The obtained results are also generalized for a certain class of estimators of $F(\lambda)$.

Full text: PDF file (2038 kB)

English version:
Theory of Probability and its Applications, 1965, 10:3, 457–465

Bibliographic databases:

Received: 14.04.1964

Citation: T. L. Malevich, “Some properties of estimators of the spectrum of a stationary process”, Teor. Veroyatnost. i Primenen., 10:3 (1965), 500–509; Theory Probab. Appl., 10:3 (1965), 457–465

Citation in format AMSBIB
\Bibitem{Mal65}
\by T.~L.~Malevich
\paper Some properties of estimators of the spectrum of a~stationary process
\jour Teor. Veroyatnost. i Primenen.
\yr 1965
\vol 10
\issue 3
\pages 500--509
\mathnet{http://mi.mathnet.ru/tvp545}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=207056}
\zmath{https://zbmath.org/?q=an:0161.15702}
\transl
\jour Theory Probab. Appl.
\yr 1965
\vol 10
\issue 3
\pages 457--465
\crossref{https://doi.org/10.1137/1110053}


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