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Solution of Shannon's problem on the monotonicity of entropy


Authors: Shiri Artstein, Keith M. Ball, Franck Barthe and Assaf Naor
Journal: J. Amer. Math. Soc. 17 (2004), 975-982
MSC (2000): Primary 94A17
DOI: https://doi.org/10.1090/S0894-0347-04-00459-X
Published electronically: May 12, 2004
MathSciNet review: 2083473
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if $X_1,X_2,\ldots$ are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum

\begin{displaymath}\mathrm{Ent}\left(\frac {X_{1}+\cdots + X_{n}}{\sqrt {n}} \right) \end{displaymath}

is an increasing function of $n$.

The result also has a version for non-identically distributed random variables or random vectors.


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Additional Information

Shiri Artstein
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Email: artst@post.tau.ac.il

Keith M. Ball
Affiliation: Department of Mathematics, University College London, Gower Street, London WC1 6BT, United Kingdom
Email: kmb@math.ucl.ac.uk

Franck Barthe
Affiliation: Institut de Mathématiques, Laboratoire de Statistique et Probabilités, CNRS UMR C5583, Université Paul Sabatier, 31062 Toulouse Cedex 4, France
Email: barthe@math.ups-tlse.fr

Assaf Naor
Affiliation: Theory Group, Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399
Email: anaor@microsoft.com

DOI: https://doi.org/10.1090/S0894-0347-04-00459-X
Keywords: Entropy growth, Fisher information, central limit theorem
Received by editor(s): September 4, 2003
Published electronically: May 12, 2004
Additional Notes: The first author was supported in part by the EU Grant HPMT-CT-2000-00037, The Minkowski Center for Geometry and the Israel Science Foundation
The second author was supported in part by NSF Grant DMS-9796221
The third author was supported in part by EPSRC Grant GR/R37210
The last author was supported in part by the BSF, Clore Foundation and EU Grant HPMT-CT-2000-00037
Article copyright: © Copyright 2004 American Mathematical Society

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