Timeline for Algebras and $\sigma$-algebras associated to random variables
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2016 at 23:01 | comment | added | Ollie | Though I do wonder if @NateEldredge's simple counterexample has a simple proof. | |
| Nov 21, 2016 at 22:56 | comment | added | Ollie | Thanks guys, this is great. The original paper by Polard jstor.org/stable/2031820?seq=1#page_scan_tab_contents on the Bernstein approximation problem also gives details of adapting his proof to $L^p$, which I'll probably peruse tomorrow - but effectively solves my original question. | |
| Nov 21, 2016 at 22:49 | comment | added | Christian Remling | @NateEldredge: My standard source for these things is Koosis, The logarithmic integral 1, and indeed I think one could finish the argument by quoting results from this book (Theorem VI D1 perhaps). In particular, in Section VI G, he assures us that essentially the same results are valid for approximation in $L^p$ as for uniform approximation (in my comment above I was concerned about this). | |
| Nov 21, 2016 at 21:33 | comment | added | Nate Eldredge | @ChristianRemling: Yeah, from some references it seems like this phenomenon is well known in the approximation theory literature. I added that tag so maybe more experts will see it. | |
| Nov 21, 2016 at 21:32 | history | edited | Nate Eldredge |
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| Nov 21, 2016 at 21:29 | comment | added | Christian Remling | @NateEldredge: This sounds like it's related to weighted polynomial approximation (though "approximation" here usually means uniform approximation), which is sometimes possible and sometimes it isn't. It would work for an exponential weight, so $\alpha<1$ is crucial here. | |
| Nov 21, 2016 at 18:08 | comment | added | Nate Eldredge | Ah, apparently it's false in general. You get a counterexample with $\Omega = [0,\infty)$, $\mathbb{P} = e^{-x^\alpha}\,dx$, $0 < \alpha < 1/2$, and $\mathcal{A}$ the polynomials; then $\alpha = \sigma$ but $\mathcal{A}$ is not dense in $L^2(\mathbb{P})$; $\sin(x)$ is not in the closure. References to references at ams.org/journals/tran/1972-168-00/S0002-9947-1972-0294655-X/… ; if I find a reference I can read I will post an answer. | |
| Nov 21, 2016 at 17:42 | comment | added | Ollie | Thanks, I'll look that up, might yield some generalisation :) - I am more interested in the unbounded case, as the Gaussian prototype suggests. | |
| Nov 21, 2016 at 16:43 | comment | added | Nate Eldredge | If your random variables were in $L^\infty$ this would follow from the multiplicative system theorem. There might be another version of it that works for unbounded functions, I'm not sure. | |
| Nov 21, 2016 at 15:25 | history | edited | Ollie | CC BY-SA 3.0 |
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| Nov 21, 2016 at 15:20 | history | edited | Ollie | CC BY-SA 3.0 |
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| Nov 21, 2016 at 15:15 | comment | added | Ollie | Edited to make these things a little clearer: but yes, the unital algebra and the minimal (complete) $\sigma$-algebra they generate. | |
| Nov 21, 2016 at 14:01 | comment | added | Nate Eldredge | When you say "the algebra they generate" do you intend to throw in the constants? I am not sure if it is strictly necessary, but I think you need it if the Wiener chaos argument is going to work. | |
| Nov 21, 2016 at 13:46 | comment | added | Ruy | I suppose the $\sigma$-algebra you refer to is the smallest sub-$\sigma$-algebra of measurable subsets of $\Omega$ relative to which all of the $\nu_\lambda$ are measurable functions. Is that right? | |
| Nov 21, 2016 at 13:02 | history | asked | Ollie | CC BY-SA 3.0 |