Timeline for A modest generalization of the law of large numbers
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 11, 2018 at 17:35 | vote | accept | Chuck Newton | ||
| Sep 10, 2018 at 23:22 | answer | added | Anthony Quas | timeline score: 2 | |
| Sep 10, 2018 at 22:21 | comment | added | Anthony Quas | I'm not aware of any prior work on this, but as I mentioned, I believe Strassen's theorem, giving conditions for the existence of monotonic couplings is likely to be useful. | |
| Sep 10, 2018 at 22:19 | comment | added | Chuck Newton | Yes, that sounds perfect. Is there any prior work on this kind of thing? (I'd never heard of Strassen's theorem, will give it a look) | |
| Sep 10, 2018 at 22:19 | comment | added | Anthony Quas | I believe the answer to my attempted reformulation of your question is that the possible pdf's of $h(X,X')$ are precisely those pdf's $g$ satisfying your conditions: $(\int_I f(x)\,dx)^2\le \int_I g(x)\,dx\le 1-(1-\int_I f(x)\,dx)^2$ for all events $I$. If this is right, the proof should probably go by Strassen's theorem. | |
| Sep 10, 2018 at 22:16 | comment | added | Anthony Quas | I suspect the question you really want to ask is the following: consider possible rules $h\colon\mathbb R^2\to\mathbb R$ where $h(x,x')\in\{x,x'\}$ (or maybe a random generalization I'll mention below). Then what are the possible pdf's of $h(X,X')$, where $X$ and $X'$ are independent samples from the pdf $f$. Here $h$ is "picking" between $x$ and $x'$. The random generalization is $h(x,x',z)\in\{x,x'\}$ where $z\in[0,1]$ is an independent Unif[0,1] random variable. | |
| Sep 10, 2018 at 21:24 | history | edited | Chuck Newton | CC BY-SA 4.0 |
added 3 characters in body
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| Sep 10, 2018 at 20:23 | history | asked | Chuck Newton | CC BY-SA 4.0 |