An entropy inequality for symmetric random variables

Jing Hao; Varun Jog

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arXiv · 1801.03868

An entropy inequality for symmetric random variables

Jing Hao·Varun Jog

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Abstract

We establish a lower bound on the entropy of weighted sums of (possibly dependent) random variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of $(X_1, X_2, \dots, X_n)$. We show that for $n \geq 3$, the lower bound is tight if and only if $X_i$'s are i.i.d.\ Gaussian random variables. For $n=2$ there are numerous other cases of equality apart from i.i.d.\ Gaussians, which we completely characterize. Going beyond sums, we also present an inequality for certain linear transformations of $(X_1, \dots, X_n)$. Our primary technical contribution lies in the analysis of the equality cases, and our approach relies on the geometry and the symmetry of the problem.

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Jing Hao, Varun Jog. 2018-01-11. An entropy inequality for symmetric random variables. https://arxiv.org/abs/1801.03868

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