  Self-similarity may originate from two origins, i.e., the process memory and
the process' increments ``infinite'' variance. A distinction is attempted by
employing the natural time \chi. Concerning the first origin, we analyze recent
data on Seismic Electric Signals, which support the view that they exhibit
infinitely ranged temporal correlations. Concerning the second, slowly driven
systems that emit bursts of various energies E obeying power-law distribution,
i.e., P(E) ~ E^-\gamma, are studied. An interrelation between the exponent
\gamma and the variance \kappa_1(= <\chi^2> - <\chi>^2) is obtained for the
shuffled (randomized) data. In the latter, the most probable value of \kappa_1
is approximately equal to that of the original data. Finally, it is found that
the differential entropy associated with the probability P(\kappa_1) maximizes
for \gamma around 1.6 to 1.7, which is comparable to the value determined
experimentally in diverse phenomena, e.g., solar flares, icequakes, dislocation
glide in stressed single crystals of ice e.t.c. It also agrees with the b-value
in the Gutenberg-Richter law of earthquakes.
