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The fractional Fisher information and the central limit theorem for stable laws. (English) Zbl 1366.60064
Summary: 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.

MSC:
60F05 Central limit and other weak theorems
26A33 Fractional derivatives and integrals
94A17 Measures of information, entropy
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