Let there be two random variables π and π with a certain joint copula. Is it always true that there is another random variable π independent from π such as the vectors $(X,Y)$ and $(X,Z)$ have the same law?
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1$\begingroup$ What do you mean by "joint copula"? "joint distribution"? $\endgroup$– Dieter KadelkaCommented Apr 16, 2020 at 9:54
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$\begingroup$ I mean that the r.v X and Y are not independent (their joint distribution is not trivial) $\endgroup$– AverroesCommented Apr 16, 2020 at 10:40
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No, suppose $X=Y$ a.s. and that they are non-degenerate. If we want $(X, Z)$ to have the same joint distribution as $(X, Y)$, we must also have $X=Z$ a.s. and hence $Y=Z$ a.s. Then $Y$ and $Z$ can obviously not be independent.
