Monotonic decrease of the non-Gaussianness of the sum of independent random variables: A simple proof

Antonio M. Tulino; Sergio Verdú

doi:10.1109/TIT.2006.880066

Monotonic decrease of the non-Gaussianness of the sum of independent random variables: A simple proof

Antonio M. Tulino, Sergio Verdú

Research output: Contribution to journal › Article › peer-review

51 Scopus citations

Abstract

Artstein, Ball, Barthe, and Naor have recently shown that the non-Gaussianness (divergence with respect to a Gaussian random variable with identical first and second moments) of the sum of independent and identically distributed (i.i.d.) random variables is monotonically nonincreasing. We give a simplified proof using the relationship between non-Gaussianness and minimum mean-square error (MMSE) in Gaussian channels. As Artstein , we also deal with the more general setting of nonidentically distributed random variables.

Original language	English (US)
Pages (from-to)	4295-4297
Number of pages	3
Journal	IEEE Transactions on Information Theory
Volume	52
Issue number	9
DOIs	https://doi.org/10.1109/TIT.2006.880066
State	Published - Sep 2006

All Science Journal Classification (ASJC) codes

Information Systems
Computer Science Applications
Library and Information Sciences

Keywords

Central limit theorem
Differential entropy
Divergence
Entropy power inequality
Minimum mean-square error (MMSE)
Non-Gaussianness
Relative entropy

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10.1109/TIT.2006.880066

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title = "Monotonic decrease of the non-Gaussianness of the sum of independent random variables: A simple proof",

abstract = "Artstein, Ball, Barthe, and Naor have recently shown that the non-Gaussianness (divergence with respect to a Gaussian random variable with identical first and second moments) of the sum of independent and identically distributed (i.i.d.) random variables is monotonically nonincreasing. We give a simplified proof using the relationship between non-Gaussianness and minimum mean-square error (MMSE) in Gaussian channels. As Artstein , we also deal with the more general setting of nonidentically distributed random variables.",

keywords = "Central limit theorem, Differential entropy, Divergence, Entropy power inequality, Minimum mean-square error (MMSE), Non-Gaussianness, Relative entropy",

author = "Tulino, {Antonio M.} and Sergio Verd{\'u}",

note = "Funding Information: Manuscript received February 13, 2006; revised May 2, 2006. This work was supported in part by the National Science Foundation under Grants NCR-0074277 and CCR-0312879. A. M. Tulino is with the Department of Electrical Engineering, Universit{\'a} di Napoli “Federico II,” Napoli, Italy, 80125 (e-mail: atulino@princeton.edu). S. Verd{\'u} is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: verdu@princeton.edu). Communicated by Y. Steinberg, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2006.880066",

year = "2006",

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doi = "10.1109/TIT.2006.880066",

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T2 - A simple proof

AU - Tulino, Antonio M.

AU - Verdú, Sergio

N1 - Funding Information: Manuscript received February 13, 2006; revised May 2, 2006. This work was supported in part by the National Science Foundation under Grants NCR-0074277 and CCR-0312879. A. M. Tulino is with the Department of Electrical Engineering, Universitá di Napoli “Federico II,” Napoli, Italy, 80125 (e-mail: atulino@princeton.edu). S. Verdú is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: verdu@princeton.edu). Communicated by Y. Steinberg, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2006.880066

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N2 - Artstein, Ball, Barthe, and Naor have recently shown that the non-Gaussianness (divergence with respect to a Gaussian random variable with identical first and second moments) of the sum of independent and identically distributed (i.i.d.) random variables is monotonically nonincreasing. We give a simplified proof using the relationship between non-Gaussianness and minimum mean-square error (MMSE) in Gaussian channels. As Artstein , we also deal with the more general setting of nonidentically distributed random variables.

AB - Artstein, Ball, Barthe, and Naor have recently shown that the non-Gaussianness (divergence with respect to a Gaussian random variable with identical first and second moments) of the sum of independent and identically distributed (i.i.d.) random variables is monotonically nonincreasing. We give a simplified proof using the relationship between non-Gaussianness and minimum mean-square error (MMSE) in Gaussian channels. As Artstein , we also deal with the more general setting of nonidentically distributed random variables.

KW - Central limit theorem

KW - Differential entropy

KW - Divergence

KW - Entropy power inequality

KW - Minimum mean-square error (MMSE)

KW - Non-Gaussianness

KW - Relative entropy

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Monotonic decrease of the non-Gaussianness of the sum of independent random variables: A simple proof

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Keywords

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