Timeline for Lower bound for the $p$-th absolute moment of a sum of random variables
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 20, 2015 at 12:40 | vote | accept | Cm7F7Bb | ||
| Feb 6, 2015 at 14:44 | comment | added | Bill Johnson | Right, VC. But the direction I mentioned does give a lot--any martingale difference sequence has lower $\ell_2$ and upper $\ell_p$ estimates when $p<2$ (and the reverse when $p>2$). This is just because martingale difference sequences are unconditional and $L_p$ has the correct type and cotype. This is all in the introductory book by Albiac and Kalton (and many other places as well). | |
| Feb 5, 2015 at 10:24 | comment | added | Cm7F7Bb | @BillJohnson Thank you very much for the comment and the reference. So it is not possible to get a better lower bound than the lower bound in the Marcinkiewicz–Zygmund inequality, right? | |
| Feb 4, 2015 at 13:47 | comment | added | Bill Johnson | Gaussians show that you do not have better than an $\ell_2$ lower bound. Mean zero independent RVs are 3-unconditional and $L_p$ has cotype 2 when $p<2$, so you indeed do have an $\ell_2$ lower estimate. See, e.g., the book by Albiac and Kalton. | |
| Feb 4, 2015 at 11:43 | comment | added | wolfies | I would suggest you change the title from 'moment' to 'fractional moment'. The term $r^{th}$ moment is conventionally taken to refer to integer values of $r$. | |
| Feb 4, 2015 at 11:32 | comment | added | Cm7F7Bb | @wolfies I'm interested in the case when $X_i$ are not necessarily identically distributed. | |
| Feb 4, 2015 at 11:27 | comment | added | wolfies | Do you wish to assume than the $X_i$ are not only independent, but also identically distributed? Because that is currently missing. | |
| Feb 4, 2015 at 10:56 | answer | added | Goulifet | timeline score: 6 | |
| Feb 4, 2015 at 10:39 | history | asked | Cm7F7Bb | CC BY-SA 3.0 |