Open AccessShannon's monotonicity problem for free and classical entropy
Author(s) -
Dimitri Shlyakhtenko,
Hanne Schultz
Publication year - 2007
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.0706451104
Subject(s) - mathematics , monotonic function , combinatorics , free probability , tuple , entropy (arrow of time) , monotone polygon , random variable , discrete mathematics , central limit theorem , physics , quantum mechanics , mathematical analysis , statistics , geometry
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. LetZ N (p )=N −1/2 (X 1 (p )+ … +X N (p )) be their central limit sums. Then the entropy (resp., free entropy) of then -tuple (Z N (1) , …,Z N (n )) is a monotone function ofN . The classical case (forn = 1) is a celebrated result of Artstein, Ball, Barthe, and Naor, and our proof is an adaptation and simplification of their argument.