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.

1$\begingroup$ taking correlation instead of covariance, do you know that there is any restriction ? Sklar's Thm (en.wikipedia.org/wiki/Copula_(probability_theory)) does not say that there is no restriction, but it does say that it is a pretty big class of ditributions. $\endgroup$– mikeCommented Feb 4, 2021 at 7:50
1 Answer
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/s4048801900915
Correlation matrices are symmetric positive semidefinite 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).
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.
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 $n≥10.$