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Teoriya Veroyatnostei i ee Primeneniya, 1964, Volume 9, Issue 2, Pages 386–390
(Mi tvp387)
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This article is cited in 8 scientific papers (total in 8 papers)
Short Communications
On the Asymptotic Behaviour of the Estimate of the Spectral Function for a Stationary Gaussian Process
T. L. Malevič Taškent
Abstract:
Let $\xi _n$, $n=0$, $\pm 1,\pm 2,\dots$, be a real stationary Gaussian sequence with an absolutely continuous spectral function $F(\lambda)$, and let $F_N (\lambda)$ be the sample spectral function.We assume that $F(\lambda)$ has no interval of constancy, and $f(\lambda)=F'(\lambda)\in L_2[0,\pi]$. Then the sequence of measures $P_N$ generated by the process $\zeta_N(\lambda)=\sqrt N[F_n(\lambda)-F(\lambda)]$ converges weakly to the measure which is generated by the Gaussian process $\zeta(\lambda)$ with ${\mathbf M}\zeta(\lambda)=0$ and
$$
{\mathbf M}\zeta(\lambda)\zeta(\mu)=2\pi\int_0^{\min(\lambda\mu)}f^2(x)\,dx.
$$
A similar result holds for the process $\xi_t$ with continuous time, $0\leqslant t<+\infty$.
Received: 22.06.1963
Citation:
T. L. Malevič, “On the Asymptotic Behaviour of the Estimate of the Spectral Function for a Stationary Gaussian Process”, Teor. Veroyatnost. i Primenen., 9:2 (1964), 386–390; Theory Probab. Appl., 9:2 (1964), 350–353
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