I am trying to understand whether it is possible and if so then how to find a joint distribution given marginals and a correlation matrix.
In particular, suppose $X_{1},\ldots,X_{n}$ are discrete random variables. I am given their marginals $p_{X_{1}},\ldots,p_{X_{n}}$ and their correlation matrix. Can I construct (uniquely?) the joint $p_{X_{1},\ldots,X_{n}}$?
I tried this for $n=2$, assuming each of the two random variables has two possible outcomes $a$ and $b>a$, and assuming that the marginals of $X_{1}$ and $X_{2}$ are the same given by $\mathbb{P}[X_{1}=a]=c$. Let $r$ be the correlation of $X_{1}$ and $X_{2}$. My Mathematica readily calculates unique joint distribution provided $c(1-r)<1$ and claims there is no joint otherwise.
How would I generalize to more random variables with more than two possible outcomes? What I do in Mathematica (calculate correlation from joint distribution and then solve for the joint distribution that gives $r$) does not generalize nicely. Also, are there some results that would tell me what I am doing can/cannot be done? Uniquely?
I tried reading StackExchange questions/answers here, here, here, here, and here. But none of the answers help me forward. Also, I know that this question is trivial when the random variables are independent. So let's disregard that special case.