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Solution of Shannon's problem on the monotonicity of entropy

Author(s): Shiri Artstein; Keith M. Ball; Franck Barthe; Assaf Naor
Journal: J. Amer. Math. Soc. 17 (2004), 975-982.
MSC (2000): Primary 94A17
Posted: May 12, 2004
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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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D. Bakry and M. Emery.
Diffusions hypercontractives.
In Séminaire de Probabilités XIX, number 1123 in Lect. Notes in Math., pages 179-206. Springer, 1985. MR 88j:60131

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K. Ball, F. Barthe, and A. Naor.
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Entropy production by block variable summation and central limit theorems.
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E. H. Lieb.
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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: 10.1090/S0894-0347-04-00459-X
PII: S 0894-0347(04)00459-X
Keywords: Entropy growth, Fisher information, central limit theorem
Received by editor(s): September 4, 2003
Posted: 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
Copyright of article: Copyright 2004, American Mathematical Society

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