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Aug 19, 2016 at 16:23 vote accept EmmGee
Aug 17, 2016 at 18:32 comment added Christian Remling @EmmGee: Your edited version runs into the exact same problem: we can take an $F$ for which $F(g_j)$ is approximately the Bernoulli RV from my example. While admittedly I haven't thought about my remark from the second paragraph much, my feeling is that it does describe what this problem is about.
Aug 17, 2016 at 7:21 comment added EmmGee But I wonder "how crucial" the extra property of rotation invariance is. Is this really necessary or can this be relaxed? I slightly reformulated the problem issue in this direction.
Aug 17, 2016 at 6:16 comment added EmmGee Thank you! This counterexample is already helpful for me because the above question is actually part of a more general statement, which is not true for $\pm 1$-values entries.
Aug 16, 2016 at 16:10 history answered Christian Remling CC BY-SA 3.0