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Costa’s concavity inequality for dependent variables based on the multivariate Gaussian copula

Part of: Communication, information Applications

Published online by Cambridge University Press:  12 April 2023

Fatemeh Asgari  and
Mohammad Hossein Alamatsaz
Fatemeh Asgari*
Affiliation:
University of Isfahan
Mohammad Hossein Alamatsaz*
Affiliation:
University of Isfahan
*
*Postal address: Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, Iran.
*Postal address: Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, Iran.
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Abstract

An extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.

Keywords

Differential entropyFisher informationGaussian copula

MSC classification

Primary: 62P30: Applications in engineering and industry
Secondary: 94A17: Measures of information, entropy 94A15: Information theory, general
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1136 - 1156
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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