  We consider the central limit theorem for stable laws in the case of the
standardized sum of independent and identically distributed random variables
with regular probability density function. By showing decay of different
entropy functionals along the sequence we prove convergence with explicit rate
in various norms to a L\'evy centered density of parameter $\lambda >1$ . This
introduces a new information-theoretic approach to the central limit theorem
for stable laws, in which the main argument is shown to be the relative
fractional Fisher information, recently introduced by the same author (cf.
arXiv 1504.07057). In particular, it is proven that, with respect to the
relative fractional Fisher information, the L\'evy density satisfies an
analogous of the logarithmic Sobolev inequality, which allows to pass from the
monotonicity and decay to zero of the relative fractional Fisher information in
the standardized sum to the decay to zero in relative entropy with an explicit
decay rate.
