Timeline for Construct a random vector as a function of another random vector
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 8, 2021 at 15:25 | comment | added | user447159 | I have posted a related question in case you could help me once again. Thanks! mathoverflow.net/questions/408022/… | |
| Nov 8, 2021 at 12:15 | comment | added | user447159 | Thanks for your help with this question. I cannot put a +1 because I don't have enough points, but I'd like to. | |
| Nov 8, 2021 at 12:14 | vote | accept | CommunityBot | ||
| Nov 6, 2021 at 23:35 | comment | added | R W | I am afraid you don't quite understand the situation. I have just explained that your Assumption 1 implies that $p_i\le 1/2$, and in my answer I explained that under this condition there is a distribution with the same $p_i$ that satisfies your conditions (1)-(3) | |
| Nov 6, 2021 at 22:31 | comment | added | R W | The argument showing that $p_i\le 1/2$ is still valid in the 3-dimensional case, and the domains used in the definition of $p_i$ are still pairwise disjoint. However, their union is not the whole space anymore (namely, the points who coordinates all have the same sign are not in this union), so that one can only claim that $\sum p_i\le 1$. | |
| Nov 6, 2021 at 20:22 | comment | added | R W | I don't quite understand what your primary goal is? Didn't you say that you want to produce distributions satisfying (1)-(3)? This is precisely what my answer is about. | |
| Nov 6, 2021 at 19:48 | comment | added | R W | Don't really know - I don't think there is a natural link between the measures on the plane and on the whole space in your setup | |
| Nov 6, 2021 at 18:54 | comment | added | user447159 | Thanks. Could you comment about the relation between Assumption 1 and your claim? In particular, are there cases where Assumption 1 holds but there are no $\pi$ measures according to your claim? | |
| Nov 6, 2021 at 18:33 | comment | added | R W | Since your marginals are symmetric, $\pi\{x<0\}=\pi\{x>0\}\ge p_1$. | |
| Nov 6, 2021 at 16:12 | comment | added | user447159 | Also, I don't get from your proof why, if $p_i>1/2$ for some $i=1,2,3$, then there CANNOT be a $\pi$ measure satisfying the desired restrictions. Which step of the proof fails? | |
| Nov 6, 2021 at 15:56 | comment | added | user447159 | Thanks. I am interested in the relation between your claim and Assumption 1: can there be cases where Assumption 1 holds but there is no $\pi$ measure according to your claim? In other words, is the assumption that each $p_i$ is less than $1/2$ also necessary and sufficient for Assumption 1 to hold? | |
| Nov 6, 2021 at 15:31 | comment | added | R W | @W13 - (1) yes; (2) I don't understand this question; (3) It is necessary and sufficient - this is precisely what the claim in bold states; (4) in addition to changing variables I have also replaced made all inequalities strict (if you wish, the measures assigning non-zero values to the boundary lines can also be dealt with in a similar way); (5) corrected | |
| Nov 6, 2021 at 15:24 | history | edited | R W | CC BY-SA 4.0 |
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| Nov 6, 2021 at 15:15 | comment | added | user447159 | (5) I think there a few typos at the end of your answer, e.g. "one can realise AND p-vector ....", which makes it hard for me to understand the conclusion of the proof. Could you check that? Thank you. | |
| Nov 6, 2021 at 15:07 | comment | added | user447159 | (4) Does your initial reformulation change the content of my question or is it just an harmless change of variables? | |
| Nov 6, 2021 at 15:04 | comment | added | user447159 | Thanks for your answer. (1) Is it correct to say that you haven't used Assumption 1 at all? (2) Is it correct to say that you have only used a condition on $p_1, p_2, p_3$ to claim your result? (3) Is the condition that each $p_i\leq 1/2$ necessary and sufficient for your claim or just sufficient? | |
| Nov 6, 2021 at 14:46 | history | answered | R W | CC BY-SA 4.0 |