Shannon's monotonicity problem for free and classical entropy
- †Department of Mathematics, University of California, Los Angeles, CA 90095; and
- §Danske Bank Ving ardssr ade 3, 1092 Copenhagen, Denmark
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Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved July 17, 2007 (received for review June 25, 2007)
Abstract
We give a short unified proof of the following theorem, valid in the context of both classical probability theory and Voiculescu's free probability theory: let (X j (1), …, X j (n)) be independent (resp., freely independent) n-tuples of random variables. Let Z N (p) = N −1/2(X 1 (p) + … + X N (p)) be their central limit sums. Then the entropy (resp., free entropy) of the n-tuple (Z N (1), …, Z N (n)) is a monotone function of N. The classical case (for n = 1) is a celebrated result of Artstein, Ball, Barthe, and Naor, and our proof is an adaptation and simplification of their argument.
Footnotes
- ‡To whom correspondence should be addressed. E-mail: shlyakht{at}math.ucla.edu
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Author contributions: D.S. and H.S. analyzed data and wrote the paper.
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The authors declare no conflict of interest.
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This article is a PNAS Direct Submission.
- Abbreviation:
- iid,
- independent identically distributed.
- © 2007 by The National Academy of Sciences of the USA
