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Teoriya Veroyatnostei i ee Primeneniya, 1965, Volume 10, Issue 3, Pages 500–509
(Mi tvp545)
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This article is cited in 3 scientific papers (total in 3 papers)
Short Communications
Some properties of estimators of the spectrum of a stationary process
T. L. Malevich Tashkent
Abstract:
Let $x_n$ ($n=0,\pm1,\pm2,\dots$) 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
$$
F_N(\lambda)=\frac1{2\pi N}\int_0^\lambda\biggl|\sum_{n=1}^Nx_ne^{-iny}\biggr|^2\,dy
$$
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 $x_n$
$$
\mathbf E\max_{0\le\lambda\le\pi}|F_N(\lambda)-F(\lambda)|^k\le C^kk!\biggl[\omega_F\biggl(\frac1N\biggr)\biggr]^{\frac k2},
$$
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 $F(\lambda)$.
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
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https://www.mathnet.ru/eng/tvp545 https://www.mathnet.ru/eng/tvp/v10/i3/p500
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