Timeline for Decoupling inequality/counterexample
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 21, 2022 at 15:48 | history | edited | Anthony Quas | CC BY-SA 4.0 |
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| Apr 21, 2022 at 15:14 | vote | accept | Aryeh Kontorovich | ||
| Apr 21, 2022 at 15:13 | comment | added | Anthony Quas | Sorry about that. The idea remains the same: $\mathbb EXY$ is an $L^2$ quantity that is much more sensitive to large changes on a small measure set than the $L^1$ quantity $\mathbb E|X-Y|$. So you can make $\mathbb E|X-Y|$ zero by making them equal. But $\mathbb E|\tilde X-\tilde Y|$ is $\Omega(1)$. Of course this doesn't give a counterexample because $\mathbb EXY$ is also $\Omega(1)$. But then you can make a perturbation that is small in $L^1$, but large in $L^2$ such that $\mathbb EXY=0$. | |
| Apr 21, 2022 at 15:09 | history | edited | Anthony Quas | CC BY-SA 4.0 |
added 126 characters in body
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| Apr 21, 2022 at 13:55 | comment | added | Aryeh Kontorovich | I guess I'll make it "unaccepted" until the proof/counterexample is found?.. | |
| Apr 21, 2022 at 13:40 | comment | added | Aryeh Kontorovich | Right I see it now. | |
| Apr 21, 2022 at 13:40 | comment | added | Iosif Pinelis | @AryehKontorovich : I think $EXY$ was OK. | |
| Apr 21, 2022 at 13:38 | comment | added | Aryeh Kontorovich | Right. Also, I'm having trouble getting $\mathbb EXY=-pM^2+(1-p)$. I'm getting | |
| Apr 21, 2022 at 13:34 | comment | added | Iosif Pinelis | Here $EX=pM\ne0$, whereas the OP requested zero means for $X$ and $Y$. Can this example be modified accordingly? | |
| Apr 21, 2022 at 13:20 | comment | added | Aryeh Kontorovich | Right, that's what I figured. Well, this puts a hamper on a certain approach. Back to the drawing board... | |
| Apr 21, 2022 at 13:17 | comment | added | Anthony Quas | I think you can scale everything without changing anything here (and truncate the normals) | |
| Apr 21, 2022 at 13:16 | comment | added | Aryeh Kontorovich | Excellent, thank you! What if we add the additional condition that all variables are in the range $[-1,1]$? | |
| Apr 21, 2022 at 13:15 | vote | accept | Aryeh Kontorovich | ||
| Apr 21, 2022 at 13:55 | |||||
| Apr 21, 2022 at 13:13 | history | answered | Anthony Quas | CC BY-SA 4.0 |