Let $X\_1, \cdots, X\_n \sim \mathrm{Unif}[0,1]$ be $n$ random variables, each with marginal distribution being a standard uniform distribution. I want to characterize the set of covariance matrices (or correlation matrix if it is easier) that can be attained by these $n$ variables. Is there a simple characterization? I feel the set might be a polyhedron.



Covariance/Correlation matrix of $n$ random variables with uniform marginal distributions
Interesting question. I am not an expert but quoting


Admissible Bernoulli correlations, by Huber and Marić
*Journal of Statistical Distributions and Applications* (2016):


<https://jsdajournal.springeropen.com/articles/10.1186/s40488-019-0091-5>



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> Correlation matrices are symmetric positive semi-definite and have all ones on the diagonal, denote this set of matrices (of size $n\times n$) as
> ${\cal E}\_n.$
> This convex compact set is called the *elliptope* (see Laurent and Poljak 1995).
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> For Gaussian marginals, the entirety of ${\cal E}\_n$ is admissible as correlations, but this is the only nontrivial set of marginals for which the question has been settled.
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> Even for other common distributions surprisingly little is known. One case that has been partially explored is that of copulas. A probability measure on $[0,1]^n$ is a copula if all its marginals are uniformly distributed on $[0,1].$ Devroye and Letac (2015) have shown that every element in ${\cal E}\_n$ is a correlation matrix for some copula, for $n≤9,$ but they believe that the statement does not hold for $n≥10.$
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