Friday, 25 November 2016, 10 a.m. (sharp), Prof. Giuseppe Toscani  (Universita' di Pavia), at the conference room of IMATI-CNR in Pavia, will give a lecture titled:

Information theoretic inequalities for log-concave and stable densities

as part of the Franco - Italian Seminar.



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

[2] G. Toscani, A concavity property for the reciprocal of Fisher
information and its consequences on Costa’s EPI, Physica A, 432 35–42

[3] G.Toscani, The fractional Fisher information and the central limit
theorem for stable laws, Ricerche Mat. 65 (1) 71—91 (2016)


Il seminario Franco-Italiano, organizzato dall'Istituto di Matematica
Applicata e Tecnologie Informatiche del CNR e dal Laboratoire Jacques
Louis Lions di Parigi, ha lo scopo di illustrare e sviluppare la
collaborazione scientifica fra ricercatori francesi e italiani nel campo
della matematica applicata.

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