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Suppose that $Z$ is a Gaussian random variable independent of $X$ and $Y$. Moreover suppose that $h(X) \geq h(Y)$, where $h(\cdot)$ is the differential Shannon entropy.

Does relation $h(X+Z) \geq h(Y+Z)$ hold in general? or under some conditions? Definitely, if they are Gaussian, it holds.

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    $\begingroup$ This claim, if true, would imply that variables with the same differential Shannon entropy necessarily also have the same Fisher information (which is essentially the derivative of entropy under the Ornstein-Uhlenbeck process), which one can readily refute with counterexamples. $\endgroup$
    – Terry Tao
    Sep 16, 2020 at 17:57
  • $\begingroup$ Thanks. Do you know any sufficient condition (except from Gaussian case) under which this holds? $\endgroup$
    – Mini
    Sep 16, 2020 at 18:00

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