Timeline for Given a total variation distance from uniform, how well can we bound the probabilities of sub-intervals?
Current License: CC BY-SA 4.0
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| Aug 29, 2021 at 20:56 | comment | added | Iosif Pinelis | @MattF. : To have a closure here, are you satisfied with this answer? | |
| Jun 14, 2021 at 13:38 | comment | added | Iosif Pinelis | A typo in Lemma 1 is fixed: $g''<0$. | |
| Jun 14, 2021 at 13:29 | comment | added | Iosif Pinelis | @AnthonyQuas : Thank you for your comment. | |
| Jun 14, 2021 at 13:29 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 14, 2021 at 5:06 | comment | added | Anthony Quas | Nice! To make this self-contained, Lemma 1 can be proved by noting that if $g$ is convex and $g(u)=0$, then $k(x)=\frac 12g(x)/(x-u)$ is increasing. The function $h$ is $\int_u^x k(x)(x-u)\big / \int_u^x (x-u)$, which is an average of an increasing function, and hence increasing. | |
| Jun 13, 2021 at 21:17 | comment | added | Iosif Pinelis | Now Mathematica does not have to be used at all. | |
| Jun 13, 2021 at 21:17 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 17:07 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 16:51 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 16:15 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 16:09 | comment | added | Iosif Pinelis | @MattF. : Such an explicit example is now given. | |
| Jun 13, 2021 at 16:09 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 15:20 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 15:19 | comment | added | user44143 | Thanks, Iosif. I deleted my previous comments. Can you give an explicit example of a distribution which attains this factor, or some factor better than $1/4$? | |
| Jun 13, 2021 at 15:14 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 15:04 | history | edited | user44143 | CC BY-SA 4.0 |
fixed typos, changed $l_1\to r$, $l_2\to s$.
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| Jun 13, 2021 at 14:27 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 14:22 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 13:07 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 7:05 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 6:58 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 13, 2021 at 6:53 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |