Let $\mathcal{G}$ be the space of all possible bivariate probability distributions.

Let's pick a bivariate probability distribution $g\in \mathcal{G}$. Can we always find a random vector $(X,Y,Z)$ such that

(1) $(X,Z)$ is distributed as $(Y,-Z)$

(2) $(Y,-Z)$ is distributed as $(-X, -Y)$

(3) $(X,Z)$ has probability distribution $g$

? As suggested in the comments below, the answer is "NO", i.e., such a random vector may not exist for some $g\in \mathcal{G}$. This remark leads to my question.


Let $\mathcal{G}^{\diamond}\subset \mathcal{G}$ be the subset of bivariate probability distributions such that a random vector $(X,Y,Z)$ satisfying (1), (2),(3) exists if and only if $g\in \mathcal{G}^{\diamond}$.

Can we list the features that a $g\in \mathcal{G}$ should have to belong to $\mathcal{G}^{\diamond}$ (necessary conditions)? For example, $g$ should be symmetric wrto zero, or $g$ should belong to a certain parametric family, etc.

Is there any result of the type "$g\in \mathcal{G}^{\diamond}$ if and only if $g$ is [XXX]" where [XXX] is filled with some non-parametric or parametric restriction?

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    $\begingroup$ I suspect there may be a critical typographical error in this question, because the claim makes no reference to $(A,B)$ and only to $\mathcal G.$ There may be at least one more, too, because the claim appears to state that $C$ and $-C$ have the same distribution, which obviously is not true for all distributions, making the question rather trivial. Could you please check your post? $\endgroup$ – whuber Nov 2 '18 at 12:46
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    $\begingroup$ You appear to be using an ambiguous notation. Could you state exactly what you mean by the tilde operator "$\sim$"? Usually it means "has the same distribution as," but as I pointed out before, with that interpretation your question is trivial, so I doubt that's what you intend it to mean. $\endgroup$ – whuber Nov 2 '18 at 13:04
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    $\begingroup$ I am afraid not. Could you describe what you are trying to say in plain English? From my experience, doing so usually brings you closer to finding the solution yourself, so could be beneficial for you as well. $\endgroup$ – Tim Nov 2 '18 at 14:49
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    $\begingroup$ I think it's clear enough: you appear to be attempting to describe, in a somewhat elliptical way, bivariate distributions with marginal distributions that are symmetric about $0.$ Unfortunately, a great deal of the formulation of this question is superfluous and therefore raises doubts concerning what you're trying to ask. For instance, there is no content at all in asserting that $(X,Y,Z)$ has a joint probability distribution: by definition, it does. The only part of the statement that appears to have any content is "$(X,Z)\sim(Y,-Z)\sim(-X,-Y).$" cc @Tim $\endgroup$ – whuber Nov 2 '18 at 15:02
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    $\begingroup$ Okay, it makes sense now. Would it be fair to interpret your question as asking how to determine when, given a random variable $(X,Z),$ there exists a random variable $Y$ satisfying the three conditions? This is somewhat interesting because although obviously $X,Y,$ and $Z$ have to be identically distributed and symmetric around zero, that is not sufficient. For example, when the correlation of $X$ and $Z$ exists and is less than $-1/2,$ such a $Y$ cannot exist. $\endgroup$ – whuber Nov 2 '18 at 15:20

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