  In the present paper the conditions for the validity of the Tsallis'
Statistics are analyzed. The same has been done following the analogy with the
traditional case: starting from the microcanonical description of the systems
and taking into account their self-similarity scaling properties in the
thermodynamic limit, it is analyzed the necessary conditions for the
equivalence of microcanonical ensemble with the Tsallis' generalization of the
canonical ensemble. It is shown that the Tsallis' Statistics is appropriate for
the macroscopic description of systems with potential scaling laws of the
asymptotic accessible states density of the microcanonical ensemble. Our
analysis shows many details of the Tsallis' formalism: the q-expectation
values, the generalized Legendre's transformations between the thermodynamic
potentials, as well as the conditions for its validity, having a priori the
possibility to estimate the value of the entropic index without the necessity
of appealing to the computational simulations or the experiment. On the other
hand, the definition of physical temperature received a modification which
differs from the Toral's result. For the case of finite systems, we have
generalized the microcanonical thermostatistics of D. H. E. Gross with the
generalization of the curvature tensor for this kind of description.
