Timeline for Upper-bound for eigenvalues of $E [UU^T]$, where $U$ is uniformly distributed on the unit $n$-sphere
Current License: CC BY-SA 4.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Apr 20, 2020 at 21:36 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 20, 2020 at 7:11 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 20, 2020 at 6:56 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 20:51 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 20:46 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 17:23 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 16:54 | comment | added | dohmatob | Oops, seems I went through all the trouble of establishing the bound (*) for nothing. Indeed, by symmetry of the sphere, for every unit vector $z \in \mathbb R^d$, the random variable $U^Tz$ has the same distribution as $U_1$ (the first coordinate of the random $n$-dimensional vector $U$). Thus, we can use the argument in mathoverflow.net/a/315232/78539 to get the (slightly more precise) bound $P(|U^Tz| > \delta) = P(|U_1| > \delta) = P(|\mathcal N(0, 1)| > \sqrt{n}\delta) + \mathcal O(1/n)$, and there are well-known tail bounds for the standard normal distribution $\mathcal N(0,1)$. | |
| Apr 19, 2020 at 15:46 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 15:40 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 15:32 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 15:02 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 14:38 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 14:15 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 13:54 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 13:48 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 12:57 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Apr 19, 2020 at 12:42 | history | asked | dohmatob | CC BY-SA 4.0 |