Timeline for Reference request: stationary measures as convex combinations of ergodic measures
Current License: CC BY-SA 3.0
15 events
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| S Sep 30, 2013 at 10:05 | history | suggested | Davide Giraudo |
added a tag.
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| Sep 30, 2013 at 10:01 | review | Suggested edits | |||
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| Sep 20, 2013 at 20:30 | comment | added | Jon Peterson | Thanks for all the useful comments. I finally found a good reference in the probability theory literature: Theorem 6.6 in "Probability Theory" by Varadhan (Courant Lecture Notes in Mathematics, volume 7). | |
| Sep 20, 2013 at 10:06 | comment | added | Ian Morris | If you're working on the metrisable topological space $[0,1]^{\mathbb{Z}}$ and looking at shift-invariant measures then all you need to do is use the Choquet-Bishop-de Leeuw Theorem on a compact convex subset of $C([0,1]^{\mathbb{Z}})^*$ with the weak-* topology, and you're done. The result is harder than this only if you're working on a completely abstract probability space, especially one which is not Lebesgue. | |
| Sep 20, 2013 at 1:18 | comment | added | R W | What's the problem? Why don't you refer to the standard results on ergodic decomposition in Lebesgue spaces? Are the spaces you are interested in non-separable? | |
| Sep 19, 2013 at 21:17 | comment | added | Anthony Quas | @Jon: I guess if you can appeal to a machine to get the regular conditional probabilities, you're left with proving ergodicity of the measure for a.e. omega. I think this is non-obvious. | |
| Sep 19, 2013 at 20:42 | comment | added | Jon Peterson | Anthony: What about the argument that I give might be more complicated than I think? I think the only sticky point in the argument is the existence of the "regular conditional probabilities" $\mu(\cdot\, | \mathcal{I})(\omega)$. However, this is always okay if the measure $\alpha$ is the distribution of a random variable on a Borel space. In the case that I'm interested in, $\alpha$ is the distribution of a random variable on the space $[0,1]^{\mathbb{Z}}$ so this should be okay. | |
| Sep 19, 2013 at 20:35 | comment | added | Jon Peterson | Asaf: It looks like the Choquet-Bishop-deLeeuw Theorem might be what I'm looking for. I'll have to check the conditions though. Typically the natural topology to work with in the space of probability measures is that of weak-* convergence. So maybe the proper locally convex topological vector space is the space of measures equipped with the weak-* topology (but of course this should maybe be checked). | |
| Sep 19, 2013 at 18:40 | comment | added | Ian Morris | Asaf: the reason for my hesitation is that I am not sure what locally convex topological vector space would be the ambient space in which the simplex of measures is embedded. | |
| Sep 19, 2013 at 18:13 | comment | added | Anthony Quas | For what it's worth, the proof of ergodic decomposition in Rudolph's book works in the generality of Lebesgue spaces | |
| Sep 19, 2013 at 18:03 | comment | added | Anthony Quas | I wrote a survey on mixing and ergodicity that appears in the (enormous) Springer Encyclopedia of Complexity. I think it's on my web page too. This has an argument along the lines that you're mentioning. I don't think it's as easy as you're saying though. You could also look at Dan Rudolph's book. Again, ergodic decomposition isn't so easy there... | |
| Sep 19, 2013 at 17:54 | comment | added | Asaf | And another remark about terminology - a stationary measure (in a Bernoulli system) is nothing more than an invariant measure. | |
| Sep 19, 2013 at 17:51 | comment | added | Asaf | Choquet's theorem works in a very general settings (std. probability space?), maybe under the name Choquet-Bishop-deLeeuw, or the "disintegration theorem". Check the little book by Phelps called "Lectures on Choquet's theorem" | |
| Sep 19, 2013 at 17:20 | comment | added | Ian Morris | In the context of ergodic versus non-ergodic invariant measures of a topological dynamical system this follows from a suitable version of Choquet's theorem, which is a general result describing how a points in a compact convex set can be written as an integral with respect to a measure which gives full measure to the set of extreme points. It is possible that something similar to this works in the abstract measurable setting but I am not sure of the details. This question may be somewhat relevant: mathoverflow.net/questions/124066/… | |
| Sep 19, 2013 at 16:29 | history | asked | Jon Peterson | CC BY-SA 3.0 |