Further investigations of Rényi entropy power inequalities and an entropic characterization of s-concave densities

Jiange Li; Arnaud Marsiglietti; James Melbourne

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arXiv · 1901.10616

Further investigations of Rényi entropy power inequalities and an entropic characterization of s-concave densities

Jiange Li·Arnaud Marsiglietti·James Melbourne

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Abstract

We investigate the role of convexity in Rényi entropy power inequalities. After proving that a general Rényi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the Rényi parameter $r\in(0,1)$, we show that random vectors with $s$-concave densities do satisfy such a Rényi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Rényi entropies of order $r\in(0,1)$ for log-concave densities and for compactly supported, spherically symmetric and unimodal densities, complementing a celebrated result of Barron (1986). Additionally, we give an entropic characterization of the class of $s$-concave densities, which extends a classical result of Cover and Zhang (1994).

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Jiange Li, Arnaud Marsiglietti, James Melbourne. 2019-09-27. Further investigations of Rényi entropy power inequalities and an entropic characterization of s-concave densities. https://arxiv.org/abs/1901.10616

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