The issue of significant features in random noise

With respect to the first example in Schimmel (2001), Van Dongen et al. (2001) conclude from their Lomb-Scargle analysis that the noise I used 'contains new periodicities that are added to the signal (these periodicities by themselves resemble a harmonic series of a 38-hour rhythm).' They infer that 'the variance of the added noise is about five times as large as the variance of the signal' causing the detection of the new significant periodicities in the noise prior to the 24-h bimodal rhythm. Moreover the 'example reflects a combination of an extremely non-sinusoidal signal with noise that is not independent, which results in a time series that is difficult to analyze with virtually any known method.' In the following, I briefly examine these concerns to avoid misunderstandings and to alert that with an inadequate use of the statistical significance test, misleading conclusions can be obtained. Although this paper further emphasizes difficulties in the detection with Lomb-Scargle periodograms, this should not be used as de-motivation. As stated in Schimmel (2001) Lomb-Scargle is a powerful technique but such as any other method one should be aware about its limitations, and use additional tools to constrain the true data characteristics.