A law of the iterated logarithm for an estimate of frequency
type: Item
instance of: scholarly article
title: A law of the iterated logarithm for an estimate of frequency
zbMATH Open document ID: 0613.60024
DOI: 10.1016/0304-4149(86)90117-1
author: E. J. Hannan
author: M. Mackisack
published in: Stochastic Processes and their Applications
publication date: 1986-01-01T00:00:00Z
review text: A single theorem is proved. This gives a form of law of the iterated logarithm for \({\hat \omega}\) an estimate of \(\omega_ 0\) in the model \[ y(t)=\rho_ 0 \cos (\omega_ 0t+\phi_ 0)+x(t). \] The process x(t) is stationary with zero mean and finite variance and is weakly mixing. The estimate \({\hat \omega}\) is the value of \(\omega\) which maximises the periodogram \(q(\omega)=| (1/T)\sum^{T}_{1}y(t)e^{it\omega}|^ 2\) and the result is that \[ \limsup_{T\to \infty}| T^{3/2}(2 \log \log T)^{-1/2}\{48\pi f(\omega_ 0)/\rho^ 2_ 0\}^{-1/2}({\hat \omega}-\omega_ 0)| =1\quad a.s. \] where f(\(\omega)\) is the spectral density of x(t). Some additional conditions on x(t) are also required.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 3992533
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: weakly mixing
zbMATH Keywords: periodogram
zbMATH Keywords: spectral density
reviewed by: Scott Duke Kominers
MaRDI profile type: Publication
full work available at URL: 0304-4149(86)90117-1
OpenAlex ID: W2057453516
cites work: The maximum of the periodogram
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description: scientific article; zbMATH DE number 3992533