Sharp variance-entropy comparison for nonnegative gaussian quadratic forms

05/24/2020

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by   Piotr Nayar, et al.

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In this article we study quadratic forms in n independent standard normal random variables. We show that among nonnegative quadratic forms, a diagonal form with equal coefficients maximizes differential entropy when variance is fixed. We also prove that differential entropy of a weighted sum of i.i.d. exponential random variables with nonnegative weights is maximized, under fixed variance, when the weights are equal.