arXiv math/05124922005-12math.OAmath.PR

熵的半循环性、高斯性和单调性

Semicircularity, Gaussianity and Monotonicity of Entropy

Hanne Schultz

备注：: As pointed out to us by Oliver Johnson, the Levy and the Cauchy distributions are both stable and have finite entropy. In the proof of Corollary 3.4 of the previous version we made a mistake by assuming that the distribution has finite variance. Corollary 3.4 has now been removed. Corollary 2.5 still holds

摘要：

S. Artstein，K.球，F。Barthe和A. Naor等证明了：若（X_j）是独立同分布的，则（X_j）是独立同分布的.随机变量，则n^{-1/2}（X_1+.+ X_n）随n的增加而增加。自由类似物最近被D. Shlyakhtenko。也就是说，如果（x_j）是非交换概率空间中自由独立、同分布的自伴元，则n^{-1/2}（x_1+.+ x_n）随着n的增加而增加。本文证明了：如果X_1（x_1，resp.）具有有限熵（分别为自由熵），如果熵（自由熵）不是n的严格递增函数，则X_1（x_1，resp.）必须为高斯分布（分别为半圆形）。

英文摘要：

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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