Timeline for Moments of complex random variables
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2020 at 11:03 | comment | added | Mateusz Wasilewski | Yes, I said that now $u$ is some distribution on the unit circle, not the uniform one. It is basically a small perturbation of the uniform distribution so that the moments are very small but positive. Could you ask more specifically, which part of the construction is still unclear? | |
| Jul 6, 2020 at 15:19 | comment | added | Farzad Aryan | On the edit: You no longer assume that $t \sim U(0, 2\pi)?$ cause that way all the moments of $u$ are zero. Also, could you please explain a bit more on how to choose $a_n$ such that $\mathbb{E} u^{k} \simeq \frac{1}{k! \mathbb{E}X^{k}}$. | |
| Jul 6, 2020 at 10:43 | history | edited | Mateusz Wasilewski | CC BY-SA 4.0 |
added 1262 characters in body
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| Jul 6, 2020 at 8:04 | comment | added | Mateusz Wasilewski | I actually realised that it's not possible to have such a fast decay of moments, even for not necessarily positive random variables. Indeed, if $X$ is a non-zero random variable with all moments finite then for some $\varepsilon>0$ the probability of the event $\{X^2>\varepsilon\}$ is positive and we have a moment bound $\mathbb{E} X^{2k} \geqslant C \varepsilon^{2k}$ which decays more slowly than the factorial. So I still don't know if there are examples satisfying your original assumptions. | |
| Jul 5, 2020 at 17:54 | comment | added | Farzad Aryan | Thanks, so basically, to put more in analysis form, an exponential sum like the one I mentioned plus an independent function which has moments like $1/k!$ would do the job. I am wondering if that is the only possibility, meaning that if we pull that oscillatory part out of $\mathcal{Z}$ we always end up with positive RV with same moments as $\mathcal{Z}.$ Could you give me an example of a RV with moments like $1/k!$? | |
| Jul 5, 2020 at 16:34 | history | answered | Mateusz Wasilewski | CC BY-SA 4.0 |