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Teoriya Veroyatnostei i ee Primeneniya, 1965, Volume 10, Issue 3, Pages 536–539
(Mi tvp550)
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Short Communications
On the second moments of an estimate of the spectral function
M. P. Shaifer Leningrad
Abstract:
A real stationary stochastic process $\{x_n\}$, $x_n=\sum_{k=-\infty}^\infty a_k\xi_{k+n}$ where $\xi_k$ are equally distributed independent random variables with $\mathbf E\xi_0=0$, $\mathbf E\xi_0^2=1$, $\mathbf E\xi_0^4<\infty$ and $\sum_{k=-\infty}^\infty a_k^2<\infty$ is considered. The asymptotic properties of the expression
$$
\operatorname{cov}\biggl(\int_{-\pi}^\pi T_1(\lambda)Y_N(\lambda)\,d\lambda,\ \int_{-\pi}^\pi T_2(\lambda)Y_N(\lambda)\,d\lambda\biggr)
$$
where
$$
Y_N(\lambda)=\frac1{2\pi N}\biggl|\sum_{j=1}^Nx_je^{i\lambda j}\biggr|^2
$$
and $\operatorname{Var}T_i(\lambda)<\infty$ ($i=1,2$) are investigated.
Received: 08.09.1964
Citation:
M. P. Shaifer, “On the second moments of an estimate of the spectral function”, Teor. Veroyatnost. i Primenen., 10:3 (1965), 536–539; Theory Probab. Appl., 10:3 (1965), 487–489
Linking options:
https://www.mathnet.ru/eng/tvp550 https://www.mathnet.ru/eng/tvp/v10/i3/p536
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Abstract page: | 156 | Full-text PDF : | 70 |
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