Timeline for Solve $\inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] $
Current License: CC BY-SA 3.0
20 events
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| Sep 13, 2017 at 8:59 | answer | added | RaphaelB4 | timeline score: 1 | |
| Sep 10, 2017 at 12:50 | comment | added | Boby | @MateuszKwaśnicki Thanks a lot for these references. | |
| Sep 10, 2017 at 12:49 | comment | added | Boby | @MTyson I am applying it to the function $\frac{1}{x}$ which is convex on $x>0$. | |
| Sep 10, 2017 at 10:40 | comment | added | Mateusz Kwaśnicki | ... Chung and Walsch's Markov Processes, Brownian Motion, and Time Symmetry (a fantastic read, but rather technical and written from a different perspective). | |
| Sep 10, 2017 at 10:40 | comment | added | Mateusz Kwaśnicki | ...If you read French, however, you should take a look into the original paper of Marcel Riesz Intégrales de Riemann–Liouville et potentiels, if only to see how the theory developed almost a century ago. A completely general theory can be found in any books on general potential theory, of which I know mostly Bliedtner and Hansen's Potential Theory An Analytic and Probabilistic Approach to Balayage (which is terrible to read, but I like the non-mathematical flavour of the title) and... | |
| Sep 10, 2017 at 10:40 | comment | added | Mateusz Kwaśnicki | @Boby: To avoid unnecessary technicalities, I suggest a book on classical potential theory. For a quick start, try John Wermer's excellent book Potential theory, which proves existence of the equilibrium measure in the first 50 pages. The case of stable processes is discussed in Landkof's book Foundations of Modern Potential Theory, but this one is relatively difficult to read. The equilibrium measure of a ball is evaluated on p. 163... | |
| Sep 10, 2017 at 1:12 | comment | added | MTyson | How are you applying Jensen's inequality to $1/(1+x^2)$? | |
| Sep 10, 2017 at 0:32 | comment | added | Boby | @MateuszKwaśnicki References would be great. I would love to learn a bit about this. | |
| Sep 9, 2017 at 21:00 | comment | added | Mateusz Kwaśnicki | ...Considering stable Lévy processes, one can get an equilibrium measure for $u(x,y)=(1-\alpha)|x-y|^{1-\alpha}$ with $\alpha\in(0,1)\cup(1,2)$ and $u(x,y)=-\log|x-y|$: these are given explicitly by beta distributions if I remember correctly, thanks to calculations from M. Riesz's 1938 paper. If you are interested, I can check the details and give some references. | |
| Sep 9, 2017 at 20:59 | comment | added | Mateusz Kwaśnicki | The "classical" potential theory tells us that the minimiser for $-\mathbb{E}|X-X'|$ is a two-point measure on the boundary. General potential theory describes more general "cost" functions: if $u(x,y)$ is a (possibly compensated) potential kernel of some Markov process (a.k.a. fundamental solution for its generator), then $\mathbb{E} u(X,X')$ is minimised by the equilibrium potential.... | |
| Sep 9, 2017 at 17:34 | history | edited | Boby | CC BY-SA 3.0 |
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| Sep 9, 2017 at 17:21 | comment | added | Christian Remling | @Boby: No, the explicit solution (from potential theory) I had in mind works for $E(-\log |X-X'|)$. | |
| Sep 9, 2017 at 17:15 | answer | added | Christian Remling | timeline score: 5 | |
| Sep 9, 2017 at 16:53 | comment | added | Boby | @ChristianRemling could you point me to some preliminary reference on equilibrium measure in potential theory? Also, are you saying that we have a solution for the case of $E[ e^{-(X-X^\prime)^2}]$? | |
| Sep 9, 2017 at 15:38 | comment | added | Mateusz Kwaśnicki | @ChristianRemling: This is very smart comment! Unfortunately, $1/(1+(x-y)^2)$ is not a potential kernel of any Lévy process (if it were a potential kernel, the generator of the corresponding Markov process would be an operator with Fourier symbol $\exp(|x|)$, which is impossible). Interestingly, even in the limiting cases $a \to \infty$ or $a \to 0^+$ there are no reasonable Markov processes behind this functional. | |
| Sep 9, 2017 at 15:34 | answer | added | Mateusz Kwaśnicki | timeline score: 3 | |
| Sep 9, 2017 at 3:12 | comment | added | Christian Remling | This is similar in spirit to finding the equilibrium measure in potential theory, except that your kernel isn't the "right" one. This analogy suggests that the optimal distribution will have the full interval as its support, with more mass sitting near the endpoints. I'm skeptical if an explicit solution is possible. In any event, you can check that $\epsilon\delta_0 +(1/2) (1-\epsilon)(\delta_a+\delta_{-a})$ beats your distribution, so that isn't optimal. | |
| Sep 8, 2017 at 22:56 | history | edited | Boby | CC BY-SA 3.0 |
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| Sep 8, 2017 at 22:43 | history | edited | Boby | CC BY-SA 3.0 |
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| Sep 8, 2017 at 22:37 | history | asked | Boby | CC BY-SA 3.0 |