  We describe how to analyze the wide class of non stationary processes with
stationary centered increments using Shannon information theory. To do so, we
use a practical viewpoint and define ersatz quantities from time-averaged
probability distributions. These ersatz versions of entropy, mutual information
and entropy rate can be estimated when only a single realization of the process
is available. We abundantly illustrate our approach by analyzing Gaussian and
non-Gaussian self-similar signals, as well as multi-fractal signals. Using
Gaussian signals allow us to check that our approach is robust in the sense
that all quantities behave as expected from analytical derivations. Using the
stationarity (independence on the integration time) of the ersatz entropy rate,
we show that this quantity is not only able to fine probe the self-similarity
of the process but also offers a new way to quantify the multi-fractality.
