Timeline for examples of random variable X independent to each of A and B, but not to (A,B)
Current License: CC BY-SA 3.0
4 events
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| Mar 19, 2013 at 15:36 | comment | added | Alexander Pruss | I've always taken a set of rvs to be independent iff every finite subset is independent. Also, I think that by the Kolmogorov 0-1 law, $Z$ isn't going to be measurable (since if $Z$ is measurable, then by symmetry $P(Z=1) = P(Z=0) = 1/2$). | |
| Sep 27, 2012 at 18:17 | comment | added | Simo_the_Wolf | Sure... By the way, also this method could be generalized to $n$ variables. It seems to be true also for countable many variables... Think of this example: take $\{ X_i \}_{i \in \mathbb{N}}$ i.i.d. variables which take values in $\mathbb{Z} / 2 \mathbb{Z}$ uniformly. Then consider $Y_n = \sum_{ i \leq n} X_i$ (where the sum is also in $\mathbb{Z} / 2 \mathbb{Z}$), and then take $Z$ as the limit of the $Y_i$s along some non-principal ultrafilter. Then $Z$ is indipendent of any $A$ proper subset of $ \{ X_i \}_{i \in \mathbb{N}}$ but clearly not indipendent from all of them. | |
| Sep 27, 2012 at 15:22 | comment | added | Sean Eberhard | ...where $S^1 = \mathbf{R}/\mathbf{Z}$ and not $\lbrace z\in\mathbf{C}: |z|=1\rbrace$. (In the latter case $X=A+B$ is not what you mean.) | |
| Sep 27, 2012 at 12:45 | history | answered | Simo_the_Wolf | CC BY-SA 3.0 |