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
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| Nov 20, 2022 at 10:11 | comment | added | Hermi | @ThomasKojar Sorry, if possible, would you mind taking a look at this question mathoverflow.net/questions/434736/…? Thank you! | |
| Nov 5, 2022 at 0:30 | comment | added | Thomas Kojar | yes indeed. But still I advise checking out their book if possible, they go into depth with many formulas. | |
| Nov 5, 2022 at 0:28 | comment | added | Thomas Kojar | thank you. Indeed, in that book the result is "iff" with continuity for the distribution. Even for atomic distributions, they have nice formulas in terms of t=0. | |
| Nov 4, 2022 at 23:42 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 4, 2022 at 23:08 | comment | added | dohmatob | Thanks for the update. Much clearer now. I see how your core argument goes through, but there are still a few of issues. First, "continuity in $t$-variable" is not for free. This will only hold if, for every $w$, the random variable $X'w$ has density. For this to hold, it is sufficient to assume that the random vector $X$ has density. Typo: Given the definition of $S_t$, I don't know what you mean by "$x_n \in S_t$". I have also posted an answer below which only assumes $X$ has density, and then uses a truncation argument (maybe this is what you had in mind in your very first comment ?). | |
| Nov 4, 2022 at 19:39 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 4, 2022 at 19:17 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 4, 2022 at 7:44 | comment | added | dohmatob | Thanks for the input (upvoted). Unfortunately, the linked book is not accessible. In particular, I don't have access to the statement of Theorem 1.7.4 (which you referenced), and so I can't tell if it provides any insight on the problem at hand. | |
| Nov 4, 2022 at 3:39 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 3, 2022 at 23:30 | history | answered | Thomas Kojar | CC BY-SA 4.0 |