Timeline for Concentration of sum of independent random variables
Current License: CC BY-SA 4.0
15 events
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| Oct 30, 2018 at 17:40 | history | edited | Davide Giraudo | CC BY-SA 4.0 |
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| May 11, 2013 at 22:16 | comment | added | Dustin G. Mixon | Related (unanswered) question: mathoverflow.net/questions/127364/… | |
| May 11, 2013 at 20:30 | comment | added | user29374 | Hi Mahdi, I thought i had a solution to your problem, but i made a silly mistake that invalidated it (hence the deleted post). However, you can easily get a lower bound for P[S_2 > t] by just making one the X_i's at least $t^{1/4}$. For sufficiently large $t$, my guess is that this will be tight up to constants - the event $\{S_2>t\}$ implies $\displaystyle \bigcup_{M} \{S > t /M \cap max X_i^2 = M\}$. You still need to overcome correlation issues, but this event should be dominated by just requiring one of the coordinates to be large. | |
| May 10, 2013 at 22:41 | comment | added | MCH | I edited the question again and rewritten sub-exponential variables as squared sub-Gaussian random variables, which is what I'm working with. I hope this resolves your issues about positivity, and makes the question more clear. | |
| May 10, 2013 at 22:38 | history | edited | MCH | CC BY-SA 3.0 |
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| May 10, 2013 at 22:26 | history | edited | MCH | CC BY-SA 3.0 |
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| May 10, 2013 at 19:19 | comment | added | cardinal | Dear Mahdi, I appreciate the edits and I am trying to be helpful, so let me encourage you to reread my first comment and take it to heart. It now appears that all three of the issues mentioned in that comment are in play here. Regarding your most recent edit and comment: (a) Your edit is incorrect and does not match Prop. 5.16 or the associated Cor. 5.17, (b) my standard normal example is entirely relevant and intended, as you will see if you examine it more carefully and (c) though somewhat immaterial thusfar, a normal random variable is most certainly subexponential (as is its square)! | |
| May 10, 2013 at 18:52 | comment | added | MCH | Sorry I had missed the absolute values in $S$ (question updated now). But that does not affect the case where the $X_i$ are non-negative. Hopefully that answers your first concern. For the second, note that normal random variables are not sub-exponential. Their second power is. | |
| May 10, 2013 at 18:51 | history | edited | MCH | CC BY-SA 3.0 |
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| May 10, 2013 at 13:07 | comment | added | cardinal | Yes, I am familiar with R. Vershynin's work; the way you gave the first bound was a dead giveaway that this was the paper you were looking at. :-) Consider the case $X_i \in \{-1,+1\}$ with probability $1/2$ each. Certainly this satisfies your conditions, but $n = S_2 \leq S^2$ is false, in general. (Take $n=2$, for instance.) As a second example, take $X_i$ to be standard normal and consider $n = 1$ to see that $\mathbb P(S_2 > t) \leq \mathbb P(S > \sqrt{t})$ is false even when $S_2 \leq S^2$. | |
| May 10, 2013 at 2:13 | comment | added | MCH | By the way, the setting is exactly the same as Proposition 5.16 on Page 14 of the following: arxiv.org/pdf/1011.3027v7.pdf | |
| May 10, 2013 at 2:12 | comment | added | MCH | I added the assumption that the $X_i$ are centered variables (zero expectation). I don't catch any other missing points. You may assume that $X_i \geq 0$ but I don't think that's necessary. Is there anything specific that doesn't make sense? | |
| May 10, 2013 at 2:08 | history | edited | MCH | CC BY-SA 3.0 |
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| May 10, 2013 at 1:48 | comment | added | cardinal | Could you please take the time to revisit your post carefully and make some edits? Several statements are incorrect. I am unsure at this point if this is due to (a) some inadvertently unstated assumptions (e.g., that the $X_i \geq 0$), (b) multiple typos or (c) something more conceptual being missed here. I don't want to make any assumptions, so I'll leave it to you to make the needed edits. Cheers. | |
| May 9, 2013 at 22:20 | history | asked | MCH | CC BY-SA 3.0 |