Non-Gaussian asymptotic minimizers in entropic uncertainty principle and the dimensional effect

Abstract : In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle and the associated cases of equality. This Shannon entropic version of the well-known Heisenberg inequality can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Rényi entropies. We recall that in both cases, equality occurs only for Gaussian random variables. However, we show that in the particular n-dimensional Laplace case, the bound is asymptotically attained as n grows. We also show numerically that this effect exists for Cauchy variables whatever the Rényi entropy considered, extending the results of Abe. These two cases are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the variable when the dimension increases, so that the effect is rather "dimensional'".
Type de document :
Communication dans un congrès
IEEE International Symposium on Information Theory (ISIT), 2006, Seattle, United States. IEEE, pp.2085-2089, 2006, Seattle, USA, 9-14 July 2006. <10.1109/ISIT.2006.261918>
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https://hal.archives-ouvertes.fr/hal-00069363
Contributeur : Steeve Zozor <>
Soumis le : mercredi 17 mai 2006 - 14:36:31
Dernière modification le : mercredi 15 avril 2015 - 16:06:35

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Steeve Zozor, Christophe Vignat. Non-Gaussian asymptotic minimizers in entropic uncertainty principle and the dimensional effect. IEEE International Symposium on Information Theory (ISIT), 2006, Seattle, United States. IEEE, pp.2085-2089, 2006, Seattle, USA, 9-14 July 2006. <10.1109/ISIT.2006.261918>. <hal-00069363>

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