Semicircularity, Gaussianity and Monotonicity of Entropy

Details Semicircularity, Gaussianity and Monotonicity of Entropy Semicircularity, Gaussianity and Monotonicity of Entropy

2005-12-21

arxiv.org/abs/math/0512492v4

Mathematics

Operator Algebras

As pointed out to us by Oliver Johnson, the Levy and the Cauchy distributions are both stable and have finite entropy. In the

Scientific paper

S. Artstein, K. Ball, F. Barthe, and A. Naor have shown that if (X_j) are i.i.d. random variables, then the entropy of n^{-1/2}(X_1+....+X_n) increases as n increases. The free analogue was recently proven by D. Shlyakhtenko. That is, if (x_j) are freely independent, identically distributed, self-adjoint elements in a noncommutative probability space, then the free entropy of n^{-1/2}(x_1+....+x_n) increases as n increases. In this paper we prove that if X_1 (x_1, resp.) has finite entropy (free entropy, resp.), and if the entropy (the free entropy, resp.) is not a strictly increasing function of n, then X_1 (x_1, resp.) must be Gaussian (semicircular, resp.).

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