Abstract
We tighten the entropy power inequality (EPI) when one of the random summands is Gaussian. Our strengthening is closely related to strong data processing for Gaussian channels and generalizes the (vector extension of) Costa's EPI. This leads to a new reverse EPI and, as a corollary, sharpens Stam's inequality relating entropy power and Fisher information. Applications to network information theory are given, including a short self-contained proof of the converse for the two-encoder quadratic Gaussian source coding problem. The proof of our main result is based on weak convergence and a doubling argument for establishing Gaussian optimality via rotational-invariance.
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