Timeline for Kolmogorov 0-1 law counter examples for almost independent variables
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 10, 2022 at 13:06 | history | edited | Will Sawin | CC BY-SA 4.0 |
deleted 4 characters in body
|
| Sep 10, 2022 at 13:06 | comment | added | Will Sawin | @PetriKattilakoski Sorry, I made an off-by-one error, it should just be $\sum_{j=0}^k (-1)^k \binom{k}{j} X_{i-j}$ so the coefficient of $Y_2 $ is $i^2 - 3 (i-1)^2 + 3 (i-2)^2 - (i-3)^2=0$ | |
| Sep 10, 2022 at 10:32 | comment | added | Petri Kattilakoski | @WillSawin Regarding (1), I fail to do so. Upon manual plugging I get for k=3, i=5 $\sum_{j=0}^{k-1}\left(-1\right)^{j}{k-1 \choose j}X_{i-j}=2Y_{2}+24Y_{3}$, for k=3,i=4 I get $2Y_{2}+18Y_{3}$ (multiplying by $k!$ won't make the $Y_2$ vanish, would it?). | |
| Aug 31, 2022 at 22:00 | comment | added | Will Sawin | @PetriKattilakoski (1) One just plugs in and simplifies. (2) I'm using a "modulo" symbol, not a "set minus" symbol. | |
| Aug 31, 2022 at 20:58 | comment | added | Petri Kattilakoski | Also, I think that a uniform distribution on $ \mathbb{R}\setminus\mathbb{Z} $ doesn't exist, does it? | |
| Aug 31, 2022 at 20:45 | comment | added | Petri Kattilakoski | Why can we write $k!Y_k$ the way you did in your last paragraph though? | |
| Aug 31, 2022 at 20:29 | comment | added | Iosif Pinelis | Thank you for your response. Do you have a reference concerning "a number of preimages equal to the absolute value of the determinant"? This is probably something simple, and yet, it would be good to see related matters in such a reference. | |
| Aug 31, 2022 at 20:21 | comment | added | Will Sawin | @IosifPinelis Sure, added two proofs, one more elementary but sketchier. | |
| Aug 31, 2022 at 20:21 | history | edited | Will Sawin | CC BY-SA 4.0 |
added 826 characters in body
|
| Aug 31, 2022 at 20:20 | comment | added | Zach Hunter | @PetriKattilakoski you are confusing quantifiers. conditioned on the outcome of $X_1,\dots$, $Y_k$ is determined by the tail sigma algebra. However, any event determined by the tail sigma algebra is constant when you condition on $X_1,\dots$. However, as Will points out, the event that $Y_k \in [0,1/2]$ is determined by the sigma algebra but occurs with probability $1/2$ (implying non-triviality). | |
| Aug 31, 2022 at 20:17 | comment | added | Will Sawin | @PetriKattilakoski First, we can't actually get the value of $Y_k$ itself using any $k$ $X_i$'s. For example adding $1/2$ to $Y_{k-1}$ and $Y_k$ does not change the value of $X_i$. Second, we absolutely can get $k! Y_k$ using $k$ of the $X_i$s, at least if they are consecutive. I didn't (intend to) say it was not constant with respect to the sigma algebra, a term I'm not familiar with. I said it was not constant, and, also, in the sigma algebra. I think by "in the sigma algebra" I mean the same thing you mean by "constant in the sigma algebra", i.e. measurable with respect to a sigma algebra. | |
| Aug 31, 2022 at 20:14 | comment | added | Iosif Pinelis | This is a very nice construction. I have only taken the liberty to make a couple of edits in your answer. Can you detail why "such maps preserve the uniform measure on the torus"? | |
| Aug 31, 2022 at 20:12 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 29 characters in body
|
| Aug 31, 2022 at 20:10 | comment | added | Petri Kattilakoski | Why isn't it constant in the tail sigma algebra? Can't we just get the value of $Y_k$ using any k $X_i$-s (which we have infinite of in the tail)? | |
| Aug 31, 2022 at 20:07 | comment | added | Will Sawin | @PetriKattilakoski I mean $Y_k$ is (1) a nonconstant function and (2) (measurable) in the tail sigma algebra, so e.g. $k! Y_k \in [0,1/2]$ is a nontrivial even in the tail sigma algebra. | |
| Aug 31, 2022 at 20:02 | comment | added | Petri Kattilakoski | What do you mean? To my understanding, the values of $Y_i$ are totally revealed given k different $X_i$-s, so specifically the values of $Y_k$ and $k!Y_k$ are revealed (and thus constant) in the tail sigma algebra of $X_i$ (?). What am I missing? | |
| Aug 31, 2022 at 19:49 | history | answered | Will Sawin | CC BY-SA 4.0 |