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Teoriya Veroyatnostei i ee Primeneniya, 1982, Volume 27, Issue 4, Pages 739–756
(Mi tvp2420)
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This article is cited in 12 scientific papers (total in 12 papers)
On the distributions of some statistical estimates of spectral density
R. Yu. Bentkus, R. A. Rudzkisa a Vilnius
Abstract:
Let $X_t$, $t=\dots,-1,0,1,\dots$, be a real Gaussian stationary time series with zero mean and spectral density $f(\lambda)$, $-\pi\le\lambda\le\pi$. In the paper the distribution of estimates (0.1) is considered, where $J_N(x)$ is the periodogram and $W\in L_1(-\pi,\pi)$. The asymptotic expansions of the distribution function and density of r. v. (0.5) are given and the theorem on large deviations is proved. Comparatively exact inequalities for the probabilities
$$
\mathbf P\{|\widehat f(\lambda)-\mathbf E\widehat f(\lambda)|\ge x\},\qquad
\mathbf P\{\|\widehat f-\mathbf E\widehat f\|_2\ge x\},\qquad
\mathbf P\{\|\widehat f-\mathbf E\widehat f\|_\infty\ge x\}
$$
are derived. It is proved also that for some of the estimates (0.1) the inequalities (3.2)–(3.4) hold for all $a>0$.
Received: 26.03.1980
Citation:
R. Yu. Bentkus, R. A. Rudzkis, “On the distributions of some statistical estimates of spectral density”, Teor. Veroyatnost. i Primenen., 27:4 (1982), 739–756; Theory Probab. Appl., 27:4 (1983), 795–814
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