Timeline for Bounding an expectation involving i.i.d. standard Gaussians and Rademacher
Current License: CC BY-SA 4.0
32 events
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| Mar 11, 2020 at 0:02 | comment | added | Clement C. | @fedja The relation between $d$ and $\gamma$ you mention is the one I'm interested in, . Why do you think this regime (especially the one for $n$) is the possibly tricky one? (Note that for the "Gaussian analogue" of the question, where $Z$ is Gaussian and a few other details change accordingly, the $-1$ become something else which blows up to infinity when $n \nearrow \gamma^{-2}$) | |
| Mar 10, 2020 at 23:54 | comment | added | Clement C. | @fedja I agree $d=1$ is not the most... interesting regime. I just don't know how to approach the problem at the moment, and simulations with $d>1$ and $n > 1$ become computationally difficult quite fast. | |
| Mar 10, 2020 at 23:52 | comment | added | fedja | For $d=1$ everything is trivially fine. The regime that bothers me is when both $d$ and $n$ are $c\gamma^{-2}$ with fixed (albeit small) $c>0$ and $\gamma\to 0$. | |
| Mar 10, 2020 at 21:34 | history | edited | Clement C. | CC BY-SA 4.0 |
Added plots for $d=1$ and $1\leq n \leq 3$.
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| Mar 9, 2020 at 7:09 | history | edited | YCor |
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| Mar 8, 2020 at 23:27 | history | edited | Clement C. | CC BY-SA 4.0 |
Removed the Gaussian case, as it turns out it's not useful for me at all after all.
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| S Mar 4, 2020 at 14:03 | history | bounty ended | CommunityBot | ||
| S Mar 4, 2020 at 14:03 | history | notice removed | CommunityBot | ||
| Mar 4, 2020 at 5:05 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Mar 4, 2020 at 4:47 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Mar 2, 2020 at 23:43 | comment | added | Clement C. | To the downvoter: is there something wrong with my question? How can I improve it? | |
| Feb 27, 2020 at 21:58 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 27, 2020 at 1:27 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 27, 2020 at 1:11 | answer | added | fedja | timeline score: 1 | |
| Feb 27, 2020 at 1:07 | answer | added | Clement C. | timeline score: 0 | |
| Feb 27, 2020 at 0:54 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 26, 2020 at 7:27 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 25, 2020 at 12:36 | comment | added | Clement C. | I can also explicitly work out the cases $n=1$ and $(n,d)=(2,1)$ to get the $\gamma^4$ dependence, unless I made a mistake in my computations. | |
| S Feb 25, 2020 at 12:20 | history | bounty started | Clement C. | ||
| S Feb 25, 2020 at 12:20 | history | notice added | Clement C. | Draw attention | |
| Feb 25, 2020 at 3:28 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 25, 2020 at 2:14 | comment | added | Clement C. | Yes, indeed (also, you're right, my above comment should read "numerator"). It's the square of the expectation (wrt $Z$). @fedja | |
| Feb 25, 2020 at 2:12 | comment | added | fedja | You mean "the numerator"? I see. $E_Z[\rm{big\ formula}]^2$ is ambiguous: I interpreted it as the expectation of the square rather than the square of the expectation. So it should be the square of the expectation in $Z$, right? | |
| Feb 25, 2020 at 1:55 | comment | added | Clement C. | @fedja (from my phone) I am not sure how you get this denominator for $n=1$. Since the square is outside of the $\mathbb{E}_Z$, the denominator should be 1 (as $\langle X,Z\rangle$ has expectation zero), shouldn't it? | |
| Feb 24, 2020 at 22:53 | comment | added | fedja | One more stupid question: when $n=d=1$ we have $E_X(\frac {E_Z\dots}{\dots})=E_X(\frac {1+\gamma^2X^2}{\dots})\ge \frac{[E_X\sqrt{1+\gamma^2X^2}]^2}{E_X\dots}=e^{-\gamma^2/2}[E_X\sqrt{1+\gamma^2X^2}]^2$, so the whole expression is at least $[E_X\sqrt{1+\gamma^2X^2}]^2-1\sim \gamma^2$. How does this agree with your conjectured $\gamma^4$ decay as $\gamma\to 0$? Am I missing or misunderstanding something? | |
| Feb 24, 2020 at 2:45 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 24, 2020 at 2:44 | comment | added | Clement C. | @fedja Oh, good catch, that was a remain of what I wrote in an earlier version for $n=2$ (I had $X,Y$ instead of $X_1,X_2$). Edited. | |
| Feb 24, 2020 at 2:41 | comment | added | fedja | Is $Y$ in $E_{XY}$ a misprint or some other r.v. $Y$ is, indeed, present? | |
| Feb 24, 2020 at 1:28 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 23, 2020 at 2:23 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 23, 2020 at 2:09 | history | edited | Clement C. | CC BY-SA 4.0 |
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| Feb 23, 2020 at 2:03 | history | asked | Clement C. | CC BY-SA 4.0 |