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

The fractional Fisher information and the central limit theorem for stable laws