Timeline for Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 4, 2022 at 7:40 | answer | added | dohmatob | timeline score: 0 | |
| Nov 4, 2022 at 3:18 | history | edited | dohmatob | CC BY-SA 4.0 |
deleted 1152 characters in body
|
| Nov 3, 2022 at 23:30 | answer | added | Thomas Kojar | timeline score: 2 | |
| Nov 2, 2022 at 22:06 | history | edited | dohmatob | CC BY-SA 4.0 |
added 50 characters in body
|
| Nov 2, 2022 at 15:55 | history | edited | dohmatob | CC BY-SA 4.0 |
added 9 characters in body
|
| Nov 2, 2022 at 11:18 | history | edited | dohmatob | CC BY-SA 4.0 |
added 36 characters in body
|
| Nov 2, 2022 at 10:15 | history | edited | dohmatob | CC BY-SA 4.0 |
added 49 characters in body
|
| Nov 2, 2022 at 9:05 | history | edited | dohmatob | CC BY-SA 4.0 |
added 67 characters in body
|
| Nov 2, 2022 at 0:47 | history | edited | dohmatob | CC BY-SA 4.0 |
deleted 10 characters in body
|
| Nov 2, 2022 at 0:28 | history | edited | dohmatob | CC BY-SA 4.0 |
added 1 character in body
|
| Nov 2, 2022 at 0:16 | history | edited | dohmatob | CC BY-SA 4.0 |
added 720 characters in body
|
| Nov 2, 2022 at 0:01 | comment | added | dohmatob | Also note that my question is sufficient conditions on $X$ to ensure that $L(S_p^n,t) \to 0$ in the limit $t\to 0^+$. So passing via Hardy-Littlewood, in the ideal case what would you expect such a condition on the density of $f$ to look like ? | |
| Nov 1, 2022 at 23:46 | comment | added | dohmatob | @ThomasKojar Thanks for bringing up Littlewood max function. However, I don't quite see how one would use this to argue about individual $u$'s that occur in the definition $L(X'w,t):=\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)$. Clear details would be appreciated. | |
| Nov 1, 2022 at 22:17 | comment | added | Thomas Kojar | how about using the maximal operator estimate (en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function) for the density f of X: $$|\{u \ : \ Proba(|X-u|<t) >t \alpha\}| \leq |\{u \ : \ (Mf)(u) > \alpha\}| \leq \frac{c}{\alpha}\int_{\mathbf{R}^n} |f|$$ to get that in in the tail ends: |u|>R for some fixed R, we have P(|X-u|<t) <t and in |u|<R we might have P(|X-u|<t) >t but because the region is bounded, we can try to apply extreme value theorem? | |
| Nov 1, 2022 at 13:25 | history | edited | dohmatob | CC BY-SA 4.0 |
added 1 character in body
|
| Nov 1, 2022 at 13:03 | history | edited | dohmatob | CC BY-SA 4.0 |
added 761 characters in body
|
| Nov 1, 2022 at 12:55 | history | edited | dohmatob | CC BY-SA 4.0 |
added 761 characters in body
|
| Nov 1, 2022 at 11:52 | comment | added | dohmatob | Any reason why this question has been downvoted without comment ? | |
| Nov 1, 2022 at 11:41 | history | edited | dohmatob | CC BY-SA 4.0 |
added 105 characters in body
|
| Nov 1, 2022 at 11:33 | history | edited | dohmatob | CC BY-SA 4.0 |
added 105 characters in body
|
| Nov 1, 2022 at 9:17 | history | edited | dohmatob | CC BY-SA 4.0 |
added 30 characters in body
|
| Nov 1, 2022 at 9:06 | history | edited | dohmatob | CC BY-SA 4.0 |
added 30 characters in body
|
| Nov 1, 2022 at 8:25 | history | edited | dohmatob | CC BY-SA 4.0 |
added 30 characters in body
|
| Nov 1, 2022 at 8:19 | history | asked | dohmatob | CC BY-SA 4.0 |