Timeline for Central Limit Theorem(s) for irrational rotation
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Feb 2, 2015 at 11:17 | answer | added | Ian Morris | timeline score: 1 | |
| Jul 1, 2014 at 13:25 | comment | added | Ian Morris | Michael T. Lacey (On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, Journal d'Analyse Mathématique 61 (1993) 47-59) investigates the precise maximum possible Hoelder exponent of a continuous function over an irrational rotation which satisfies a functional CLT, giving this maximum in terms of the irrationality measure of the rotation number. Perhaps this is of interest. | |
| Jul 1, 2014 at 7:30 | answer | added | Robert Israel | timeline score: 0 | |
| Jun 30, 2014 at 19:45 | comment | added | Christian Remling | @MarcinKotowski I am aware of that. What I was trying to say was that a circle rotation is as non-random as possible (given that it's ergodic); for example, it has pure point spectrum. Of course, I said it in a maximally misleading way. | |
| Jun 30, 2014 at 19:01 | vote | accept | Marcin Kotowski | ||
| Jun 30, 2014 at 15:55 | answer | added | Andreas Thom | timeline score: 21 | |
| Jun 30, 2014 at 14:04 | history | edited | Noah Stein |
Adding ergodic theory tag
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| Jun 30, 2014 at 10:28 | comment | added | Marcin Kotowski | @Christian: CLT does hold for dynamical systems which are sufficiently ergodic (e.g. hyperbolic actions on a torus). So independence is far from necessary for CLT to hold. | |
| Jun 30, 2014 at 1:00 | comment | added | Anthony Quas | There is a paper of Harry Kesten (Uniform Distribution Mod 1) published in the Annals in 1960 -- also a follow-up paper a couple of years later, dealing with the case where $f$ is a characteristic function minus its expectation, and showing the limit distribution is Cauchy | |
| Jun 30, 2014 at 0:51 | comment | added | Christian Remling | Maybe you're more looking for something along the lines of "error estimates in the ergodic theorem." See for example this question: mathoverflow.net/questions/4411/… | |
| Jun 30, 2014 at 0:44 | comment | added | Christian Remling | Wouldn't anything resembling the CLT be a miracle in this situation? After all, the summands are anything but independent. | |
| Jun 29, 2014 at 21:13 | comment | added | Marcin Kotowski | @DouglasZare: edited so it's clear the relevant r.v. have mean zero. | |
| Jun 29, 2014 at 21:12 | history | edited | Marcin Kotowski | CC BY-SA 3.0 |
added 70 characters in body
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| Jun 29, 2014 at 20:59 | comment | added | Douglas Zare | What sort of conditions do you impose on $f$? You mention smooth functions but your example is not smooth. It looks like you are assuming that $\int_{S^1} f(z) ds = 0$. | |
| Jun 29, 2014 at 20:16 | history | asked | Marcin Kotowski | CC BY-SA 3.0 |