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Information theoretic inequalities for log-concave and stable densities
Prof. Giuseppe Toscani, Universitą di Pavia
FRANCO-ITALIAN SEMINAR, Sala Conferenze, IMATI-CNR - Venerdì 25 Novembre 2016 h.10:00
Abstract. Some new information inequalities are presented.
In the first part, we show that Shannon’s entropy-power inequality for probability densities admits a strengthened version in the case in which the densities are log-concave. In such a case, in fact, one can extend the Blachman–Stam argument to obtain a sharp inequality for the second derivative of Shannon’s entropy functional with respect to the heat semigroup [1]. As a byproduct one proves that the third derivative of the entropy power of a log-concave probability density is nonnegative in time with respect to the addition of a Gaussian noise [2]. For log-concave densities this improves the well-known Costa’s concavity property of the entropy power.
In the second part, we introduce the new concept of fractional relative Fisher information, by showing that it satisfies an analogous of the Blachman–Stam inequality. This allows to prove new convergence results to the Lévy density in the central limit theorem for stable laws [3].
[1] G. Toscani, A strengthened entropy power inequality for log-concave densities. IEEE Transactions on Information Theory 61 (12) 6550–6559 (2015)
[2] G. Toscani, A concavity property for the reciprocal of Fisher information and its consequences on Costa’s EPI, Physica A, 432 35–42 (2015)
[3] G.Toscani, The fractional Fisher information and the central limit theorem for stable laws, Ricerche Mat. 65 (1) 71—91 (2016)
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