  The notion of fractality, in the context of positive-valued probability
distributions, is conventionally associated with the class of Paretian
probability laws. In this research we show that the Paretian class is merely
one out of six classes of probability laws - all equally entitled to be
ordained fractal, all possessing a characteristic power-law structure, and all
being the unique fixed points of renormalizations acting on the space of
positive-valued probability distributions. These six fractal classes are
further shown to be one-dimensional functional projections of underlying
fractal Poisson processes governed by: (i) a common elemental power-law
structure; and, (ii) an intrinsic scale which can be either linear, harmonic,
log-linear, or log-harmonic. This research provides a panoramic and
comprehensive view of fractal distributions, backed by a unified theory of
their underlying Poissonian fractals.
