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

 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:

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}