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Teor. Veroyatnost. i Primenen., 1965, Volume 10, Issue 3, Pages 536–539 (Mi tvp550)  

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
$$ \operatorname{cov}(\int_{-\pi}^\pi T_1(\lambda)Y_N(\lambda) d\lambda, \int_{-\pi}^\pi T_2(\lambda)Y_N(\lambda) d\lambda) $$
where
$$ Y_N(\lambda)=\frac1{2\pi N}|\sum_{j=1}^Nx_je^{i\lambda j}|^2 $$
and $\operatorname{Var}T_i(\lambda)<\infty$ ($i=1,2$) are investigated.

Full text: PDF file (845 kB)

English version:
Theory of Probability and its Applications, 1965, 10:3, 487–489

Bibliographic databases:

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

Citation in format AMSBIB
\Bibitem{Sha65}
\by M.~P.~Shaifer
\paper On the second moments of an estimate of the spectral function
\jour Teor. Veroyatnost. i Primenen.
\yr 1965
\vol 10
\issue 3
\pages 536--539
\mathnet{http://mi.mathnet.ru/tvp550}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=189132}
\zmath{https://zbmath.org/?q=an:0168.17302}
\transl
\jour Theory Probab. Appl.
\yr 1965
\vol 10
\issue 3
\pages 487--489
\crossref{https://doi.org/10.1137/1110058}


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