%0 Journal Article %1 hao2018entropy %A Hao, Jing %A Jog, Varun %D 2018 %K entropy-inequalities %T An entropy inequality for symmetric random variables %U http://arxiv.org/abs/1801.03868 %X 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 $n 3$, 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.

@article{hao2018entropy, 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.}, added-at = {2018-01-17T23:05:26.000+0100}, author = {Hao, Jing and Jog, Varun}, biburl = {https://www.bibsonomy.org/bibtex/2e76de3c6bdd0588ed6852043ee4a72e4/claired}, description = {An entropy inequality for symmetric random variables}, interhash = {92ca335173cfb299c48d21a2e77dc081}, intrahash = {e76de3c6bdd0588ed6852043ee4a72e4}, keywords = {entropy-inequalities}, note = {cite arxiv:1801.03868Comment: submitted to ISIT 2018}, timestamp = {2018-01-17T23:05:26.000+0100}, title = {An entropy inequality for symmetric random variables}, url = {http://arxiv.org/abs/1801.03868}, year = 2018 }