  Standard shot-noise models, which seek to explain the broadband noise
variability that characterises the X-ray lightcurves of X-ray binaries and
active galaxies, predict that the power spectrum of the X-ray lightcurve is
stationary (i.e. constant amplitude and shape) on short time-scales. We show
that the broadband noise power spectra of the black hole candidate Cyg X-1 and
the accreting millisecond pulsar SAX J1808.4-3658 are intrinsically
non-stationary, in that RMS variability scales linearly with flux.
Flux-selected power spectra confirm that this effect is due to changes in
power-spectral amplitude and not shape. The lightcurves of three Seyfert
galaxies are also consistent with a linear relationship between RMS variability
and flux, suggesting that it is an intrinsic feature of the broadband noise
variability in compact accreting systems over more than 6 decades of central
object mass. The RMS variability responds to flux variations on all measured
time-scales, raising fundamental difficulties for shot-noise models which seek
to explain this result by invoking variations in the shot parameters. We
suggest that models should be explored where the longest time-scale variations
are fundamental and precede the variations on shorter time-scales. Possible
models which can explain the linear RMS-flux relation include the fractal
break-up of large coronal flares, or the propagation of fluctuations in mass
accretion rate through the accretion disk. The linear relationship between RMS
variability and flux in Cyg X-1 and SAX J1808.4-3658 is offset on the flux
axis, suggesting the presence of a second, constant-flux component to the
lightcurve which contributes 25% of the total flux. The spectrum of this
constant component is similar to the total spectrum [abridged].
