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Gaussian functions / distributions / processes...
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Comparison of Rademacher and Gaussian expected values under linear transformations
At the end I'm not sure if you want $G$ to be a standard Gaussian or a Rademacher? … If $G$ were to be a standard Gaussian, then take $U=I_n$ and consider the function $f$ such that $f(1)=f(-1)=1$ and $f(x)=0$ for all $x\not\in \{1,-1\}$. …