  The aim of this paper is to shed light on the analysis of non-stationary time
series by means of the method of diffusion entropy. For this purpose, we first
study the case when infinitely many time series, as different realizations of
the same dynamic process, are available, so as to adopt the Gibbs ensemble
perspective. We solve the problem of establishing under which conditions
scaling emerges from within this perspective. Then, we study the more
challenging problem of creating a diffusion process from only one single
(non-stationary) time series. The conversion of this single sequence into many
diffusional trajectories is equivalent to creating a non-Gibbsian ensemble.
However, adopting a probabilistic approach to evaluate the contribution of any
system of this non-Gibbsian ensemble, and using for it the theoretical Gibbsian
prescription of the earlier case, we find a recipe that fits accurately the
numerical results. With the help of this recipe we show that nonstationary time
series produce either anomalous scaling with ordinary statistics or ordinary
scaling with anomalous statistics. From this recipe we also derive an
attractive way to explain the entropy time evolution, as resulting from two
distinct uncertainty sources, the lack of information on the trajectory initial
condition, and the lack of control on random trajectories.
