  A new information-theoretic approach to the central limit theorem for stable
laws is presented. The main novelty is the concept of relative fractional
Fisher information, which shares most of the properties of the classical one,
included Blachman-Stam type inequalities. These inequalities relate the
fractional Fisher information of the sum of $n$ independent random variables to
the information contained in sums over subsets containing $n-1$ of the random
variables. As a consequence, a simple proof of the monotonicity of the relative
fractional Fisher information in central limit theorems for stable law is
obtained, together with an explicit decay rate.
