This question is a step in answering this question on the stats.se.
Given a distribution $F(X_1,\ldots,X_n)$ on the nonnegative orthant $\mathbb{R}_+^n$ (i.e. each of the marginals is supported on the nonnegative reals). Where the mean of each marginal is 1 (i.e. $E(X_i)=1$ for all $i$). What are the restrictions on the covariance matrix (assuming that it exists, other than positive semi-definiteness)?
The idea is to be able to recognize a covariance matrix as coming from a nonegative multivariate distribution. For example $\pmatrix{4&-3\\-3& 4}$ is a perfectly fine covariance matrix, it is symmetric and positive definite, but it cannot come from a non-negative multivariate ditribution with mean $\mathbf 1$ because $\text{Cov}(X_1,X_2)=E(X_1X_2)-1\ge-1$ as $E(X_1X_2)$ is positive. I am certain that this is not the only such restriction.