## The Annals of Probability

### A Stability Result for the Periodogram

#### Abstract

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 $\delta > 0$.

#### Article information

Source
Ann. Probab., Volume 18, Number 4 (1990), 1765-1783.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990647

Digital Object Identifier
doi:10.1214/aop/1176990647

Mathematical Reviews number (MathSciNet)
MR1071824

Zentralblatt MATH identifier
0716.62092

JSTOR