Timeline for Convex combinations of Bernoulli Measures
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2016 at 17:17 | answer | added | Dominik Kwietniak | timeline score: 4 | |
| Feb 25, 2016 at 16:01 | comment | added | Ian Morris | Sorry, "where $\mathbb{P}$ is the weak-* limit of the measures $\mathbb{P}_n$". | |
| Feb 25, 2016 at 15:39 | comment | added | Ian Morris | I don't see anything complicated about this limit. Write each term in the sequence as $\int m\,d\mathbb{P}_n$ where $\mathbb{P}_n=\sum_{i=1}^{M_n} \alpha_i^n \delta_{Ber_i^n}$, then $\eta = \int m\,d\mathbb{P}$ where $\mathbb{P}$ is the weak-* limit of the measures $m_n$. If $\mathbb{P}$ is a point mass then this is a Bernoulli measure, if not then $\eta$ is not ergodic. | |
| Feb 25, 2016 at 15:28 | comment | added | Bruno Brogni Uggioni | Dear @IanMorris, every measure in $C$ is a proper linear combination of invariant measures, but, in the closure of $C$, for instance the $\eta$ I wrote before: $$\eta = \lim_{n}\sum_{i=1}^{M_{n}}\alpha_{i}^{n}Ber_{i}^{n},$$ you may have $M_{n}\rightarrow \infty$, so, it looks like you can have complicated measures on the closure... | |
| Feb 25, 2016 at 13:56 | comment | added | Ian Morris | If an invariant measure is a proper linear combination of two non-identical invariant measures, it is not ergodic. This is Theorem 6.10.iii in Walters' book. Every measure $\eta$ in the closure of $C$ is a proper linear combination of invariant measures unless the associated measure $\mathbb{P}$ is a Dirac measure, in which case $\eta$ is a Bernoulli measure. | |
| Feb 25, 2016 at 13:00 | comment | added | Bruno Brogni Uggioni | Dear @NoahStein, yes, professor Ian Morris is right, each $i$ refers to a Bernoulli measure $\eta_{i}$ | |
| Feb 25, 2016 at 12:54 | comment | added | Bruno Brogni Uggioni | Dear @IanMorris, are you sure that the only ergodic measures on that space are Bernoulli measures? It means that if a ergodic measure $\eta$ satisfies: $$\eta = \lim_{n}\sum_{i=1}^{M_{n}}\alpha_{i}^{n}Ber_{i}^{n}$$ so it must be Bernoulli... well, I will try more, but I can't see this even when I suppose $\eta$ mixing... thanks for your attention | |
| Feb 24, 2016 at 23:25 | vote | accept | Bruno Brogni Uggioni | ||
| Feb 24, 2016 at 19:00 | history | edited | Michael Hardy | CC BY-SA 3.0 |
added 8 characters in body
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| Feb 24, 2016 at 18:47 | comment | added | Ian Morris | Noah, I'm not sure I understand: doesn't $\eta_i$ satisfy $\eta_i([x_1\cdots x_n])=p^i_{x_1}\cdots p^i_{x_n}$ for each $x_1,\ldots,x_n$? | |
| Feb 24, 2016 at 18:34 | comment | added | Noah Stein | Could you clarify whether you actually care about invariantness, or where you meant to put it? You mention it, but then use distinct $p^i_j$ for different $j$ which is not invariant. | |
| Feb 24, 2016 at 18:28 | answer | added | Ian Morris | timeline score: 3 | |
| Feb 24, 2016 at 18:14 | comment | added | Ian Morris | I'll expand on this as an answer. | |
| Feb 24, 2016 at 18:09 | comment | added | Bruno Brogni Uggioni | @IanMorris, sorry, but $\int m d\mathbb{P}(m)$ is kind of a ergodic decomposition? I mean, a Borel probability on the space of Bernoulli measures... I don't understand how this integral you wrote works... | |
| Feb 24, 2016 at 18:02 | comment | added | Ian Morris | This is precisely the set of all measures of the form $\int m\, d\mathbb{P}(m)$ where $\mathbb{P}$ is a Borel probability measure on the set of Bernoulli measures (a closed set). So, it's a closed face of the set of invariant measures. | |
| Feb 24, 2016 at 18:01 | comment | added | Bruno Brogni Uggioni | @Algernon "Bigness" in the sense this set can be almost everything or, on the opposite, has so many properties, is so specific that only a few invariant measures can be there... | |
| Feb 24, 2016 at 17:58 | comment | added | Algernon | What is your measure of "bigness"? de Finetti's theorem states that the convex mixtures of Bernoulli measures are precisely the measures that are invariant under finite permutations of the indices. | |
| Feb 24, 2016 at 17:48 | history | asked | Bruno Brogni Uggioni | CC BY-SA 3.0 |