An Entropy Inequality for Symmetric Random Variables

We establish a lower bound on the entropy of weighted sums of (possibly dependent) random variables (X1, X2, ..., X n) possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of (X1, X2, ..., Xn). We show that for n ≥ 3, the lower bound is tight if and only if Xi'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 (X1, ..., 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.