On the use of structure functions to study blazar variability: caveats
  and problems
  The extensive use of the structure function (SF) in the field of blazar
variability suggests that characteristics time-scales are embedded in the light
curves of these objects. We argue that for blazar variability studies, the SF
results are sometimes erroneously interpreted leading to misconceptions about
the actual source properties. Based on extensive simulations we caution that
spurious breaks will appear in the SFs of almost all light curves, even though
these light curves may contain no intrinsic characteristic time-scales. i.e.
having a featureless underlying power-spectral-density (PSD). We show that the
time-scales of the spurious SF-breaks depend mainly on the length of the
artificial data set and also on the character of the variability i.e. the shape
of the PSD. The SF is often invoked in the framework of shot-noise models to
determine the temporal properties of individual shots. We caution that although
the SF may be fitted to infer the shot parameters, the resultant shot-noise
model is usually inconsistent with the observed PSD. As any model should fit
the data in both the time and the frequency domain the shot-noise model, in
these particular cases, can not be valid. Moreover, we show that the lack of
statistical independence between adjacent SF points, in the standard SF
formulation, means that it is not possible to perform robust statistical model
fitting following the commonly used least-squares fitting methodology. The
latter yields uncertainties in the fitting parameters (i.e. slopes, breaks)
that are far too small with respect to their true statistical scatter. Finally,
it is also commonly thought that SFs are immune to the sampling problems, such
as data gaps, which affects the estimators of the PSDs. However we show that
SFs are also troubled by gaps which can induce artefacts.
