Timeline for De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 21 at 20:29 | comment | added | Andrei Kh | I know this is 12 years later, but is there any direct reference for (B)? | |
| Jul 14, 2013 at 13:27 | answer | added | Jason Rute | timeline score: 5 | |
| Apr 11, 2012 at 6:02 | answer | added | Thierry de la Rue | timeline score: 4 | |
| Apr 10, 2012 at 23:36 | comment | added | James Martin | Agreed - works fine with open balls. | |
| Apr 10, 2012 at 21:44 | history | edited | Jason Rute | CC BY-SA 3.0 |
Changed "measurable sets" to "open balls", and "\in" to "\subseteq"
|
| Apr 10, 2012 at 21:40 | comment | added | Jason Rute |
@James Martin: Ok, I see your point. I could take $A={x_0,x_1, \ldots}$. I got this from a paper, which in turn, got it from Kallenberg, Probabilistic symmetries and invariance principles, Proposition 1.4: books.google.com/… I must be reading the a.s. in that statement incorrectly. I think it works if I assume $A$ ranges over all open balls. (I could also use continuous functions like equation (B)).
|
|
| Apr 10, 2012 at 21:01 | history | edited | Jason Rute | CC BY-SA 3.0 |
Fixed typo
|
| Apr 10, 2012 at 20:31 | history | edited | Jason Rute | CC BY-SA 3.0 |
Fixed typos
|
| Apr 10, 2012 at 20:29 | comment | added | Jason Rute | @James Martin: I think it is correct. This isn't true for all x, but for a.e. x. In other words, there are $x$'s so random that the $x_i$ are "nicely" distributed in a way that gives the measure $\nu_x$. For example, if the $x_i$ were uniformly distributed, then (A) would hold for the $\nu_x$ equal to the Lebesgue measure. This is similar to the concept of being a generic in ergodic theory (see terrytao.wordpress.com/2008/02/04/254a-lecture-9-ergodicity). Let me know if you still disagree. | |
| Apr 10, 2012 at 20:08 | comment | added | James Martin | It seems there's something wrong with the order of the quantifiers in 2. For instance, if the marginal distribution of $X_1$ is non-atomic, then the limit in (A) won't exist for all $A$ (if $A$ is allowed to depend on $x$, then $A$ could for example pick out certain individual values of the $x_i$ but not others in an inconvenient way). Do you mean: for all $A$, then for $\mu$-a.e. $x$, ...? | |
| Apr 10, 2012 at 18:54 | history | asked | Jason Rute | CC BY-SA 3.0 |