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October, 1990 A Stability Result for the Periodogram
K. F. Turkman, A. M. Walker
Ann. Probab. 18(4): 1765-1783 (October, 1990). DOI: 10.1214/aop/1176990647


Let $\{X_t\}^\infty_{t=1}$ be a stationary Gaussian time series with zero mean, unit variance, absolutely summable autocorrelation function and at least once differentiable spectral density function which is strictly positive in $\lbrack 0, \pi \rbrack$. In this paper it is shown that, if $M_n$ denotes the maximum of the normalized periodogram of $\{X_1,\ldots, X_n\}$ over the interval $\lbrack 0, \pi \rbrack$, then, almost surely, \begin{equation*}\tag{1} \lim \inf_{n\rightarrow\infty} \lbrack M_n - 2 \log n + \log \log n \rbrack \geq 0\end{equation*} and \begin{equation*}\tag{2} \lim \sup_{n\rightarrow\infty} \lbrack M_n - 2 \log n - 2(\log n)^\delta \rbrack = -\infty\end{equation*} for any $\delta > 0$.


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K. F. Turkman. A. M. Walker. "A Stability Result for the Periodogram." Ann. Probab. 18 (4) 1765 - 1783, October, 1990.


Published: October, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0716.62092
MathSciNet: MR1071824
Digital Object Identifier: 10.1214/aop/1176990647

Primary: 60F15
Secondary: 62F15

Keywords: periodogram , spectral density function , trigonometric polynomials

Rights: Copyright © 1990 Institute of Mathematical Statistics

Vol.18 • No. 4 • October, 1990
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