Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities

Jiange Li; Arnaud Marsiglietti; James Melbourne

doi:10.1007/978-3-030-46762-3_4

Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities

Jiange Li
, Arnaud Marsiglietti^*
, James Melbourne

^*Corresponding author for this work

Hebrew University of Jerusalem
University of Florida
University of Minnesota Twin Cities

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

11 Scopus citations

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 and Chistyakov (IEEE Trans Inform Theory 61(2):708–714, 2015) fails when the Rényi parameter r ∈ (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 ∈ (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).

Original language	English
Title of host publication	Lecture Notes in Mathematics
Publisher	Springer
Pages	95-123
Number of pages	29
DOIs	https://doi.org/10.1007/978-3-030-46762-3_4
State	Published - 2020
Externally published	Yes

Publication series

Name	Lecture Notes in Mathematics
Volume	2266
ISSN (Print)	0075-8434
ISSN (Electronic)	1617-9692

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10.1007/978-3-030-46762-3_4

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title = "Further Investigations of R{\'e}nyi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities",

abstract = "We investigate the role of convexity in R{\'e}nyi entropy power inequalities. After proving that a general R{\'e}nyi entropy power inequality in the style of Bobkov and Chistyakov (IEEE Trans Inform Theory 61(2):708–714, 2015) fails when the R{\'e}nyi parameter r ∈ (0, 1), we show that random vectors with s-concave densities do satisfy such a R{\'e}nyi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for R{\'e}nyi 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).",

author = "Jiange Li and Arnaud Marsiglietti and James Melbourne",

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AU - Marsiglietti, Arnaud

AU - Melbourne, James

PY - 2020

Y1 - 2020

N2 - 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 and Chistyakov (IEEE Trans Inform Theory 61(2):708–714, 2015) fails when the Rényi parameter r ∈ (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 ∈ (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).

AB - 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 and Chistyakov (IEEE Trans Inform Theory 61(2):708–714, 2015) fails when the Rényi parameter r ∈ (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 ∈ (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).

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Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities

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