Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities
Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities
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Rényi 熵幂不等式的进一步研究和 s 凹密度的熵表征
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发表时间：
2019
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通讯作者：
J. Melbourne
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作者：
Jiange Li;Arnaud Marsiglietti;J. Melbourne
研究了凸性在Renyi熵幂不等式中的作用。在证明了Bobkov和Chistyakov（IEEE Trans Inform Theory 61（2）：708-714，2015）风格的一般Renyi熵幂不等式在Renyi参数r ∈（0，1）时失效之后，我们证明了具有s-凹密度的随机向量确实满足这样的Renyi熵幂不等式。沿着这条路，我们建立了对数凹密度和紧支撑的球对称单峰密度的r ∈（0，1）阶Renyi熵的中心极限定理的收敛性，补充了巴伦（Ann Probab 14：336-342，1986）的一个著名结果。另外，我们给出了s-凹密度类的熵刻画，推广了Cover和Zhang（IEEE Trans Inform Theory 40（4）：1244-1246，1994）的经典结果.
We investigate the role of convexity in Renyi entropy power inequalities. After proving that a general Renyi entropy power inequality in the style of Bobkov and Chistyakov (IEEE Trans Inform Theory 61(2):708–714, 2015) fails when the Renyi parameter r ∈ (0, 1), we show that random vectors with s-concave densities do satisfy such a Renyi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Renyi entropies of order r ∈ (0, 1) for log-concave densities and for compactly supported, spherically symmetric and unimodal densities, complementing a celebrated result of Barron (Ann Probab 14:336–342, 1986). Additionally, we give an entropic characterization of the class of s-concave densities, which extends a classical result of Cover and Zhang (IEEE Trans Inform Theory 40(4):1244–1246, 1994).