  We demonstrate a simple method for testing the significance of peaks in the
periodogram of red noise data. The procedure was designed to test for spurious
periodicities in X-ray light curves of active galaxies, but can be used quite
generally to test for periodic components against a background noise spectrum
assumed to have a power law shape. The method provides a simple and fast test
of the significance of candidate periodic signals in short, well-sampled time
series such as those obtained from XMM-Newton observations of Seyfert galaxies,
without the need for Monte Carlo simulations. A full account is made of the
number of trials and the uncertainties inherent to the model fitting. Ignoring
these subtle effects can lead to substantially overestimated significances.
These difficulties motivate us to demand high standards of detection (minimum
>99.9 per cent confidence) for periodicities in sources that normally show red
noise spectra. The method also provides a simple means to estimate the power
spectral index, which may be an interesting parameter itself, regardless of the
presence/absence of periodicities.
