  We use both time-domain and Fourier techniques to study the correlated
optical and X-rays variability in the black hole X-ray nova XTE J1118+480. Its
X-ray timing properties, such as the shape of the X-ray power spectrum and
cross-correlation functions (CCFs) between X-ray energy bands, the slight
decrease of RMS variability from 30 % in the 2-5.9 keV to 19 % in the
15.5--44.4 keV band, as well as the X-ray hardness/flux anti-correlation, are
very similar to what is found in other black hole binaries in the hard state.
The optical/X-ray CCF is virtually independent of the X-ray energies. The
optical flux appears to be correlated not only with the X-ray flux but also
with the X-ray spectral variability. Both the coherence function and the lags
between optical and the X-rays are Fourier frequency dependent. The
optical/X-ray coherence function reaches its maximum (~0.3) in the 0.1-1 Hz
range and the time-lags decrease with frequency approximatively like f^-0.8.
The correlation between X-ray and optical light curves appears to have
time-scale-invariant properties. The X-ray/optical CCF maintains a similar but
rescaled shape on time-scales ranging at least from 0.1 s to few 10 s. Using
the event superposition method we show that the correlation is not triggered by
a single type of event (dip or flare) in the light curves. Instead, optical and
X-ray fluctuations of very different shapes, amplitudes and time-scales are
correlated in a similar mode where the optical light curve is seemingly related
to the time derivative of the X-rays.
