What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? (solved)

Paulo (from the Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil) has posted an answer to my earlier question both as a comment on the ‘Og and as a solution on StackOverflow (with a much more readable LaTeX output). His solution is based on the observation that the multidimensional log-normal distribution still allows for closed form expressions of both the mean and the variance and that those expressions can further be inverted to impose the pair (μ,Σ)  on the log-normal vector. In addition, he shows that the only constraint on the covariance matrix is that the covariance σ_ij is larger than -μ_iμ_j.. Very neat!

In the meanwhile, I corrected my earlier R code on the gamma model, thanks to David Epstein pointing a mistake in the resolution of the moment equation and I added the constraint on the covariance, already noticed by David in his question. Here is the full code:

sol=function(mu,sigma){
  solub=TRUE
  alpha=rep(0,3)
  beta=rep(0,2)
  beta[1]=mu[1]/sigma[1]
  alpha[1]=mu[1]*beta[1]
  coef=mu[2]*sigma[1]-mu[1]*sigma[3]
  if (coef<0){
   solub=FALSE}else{
    beta[2]=coef/(sigma[1]*sigma[2]-sigma[3]^2)
    alpha[2]=sigma[3]*beta[2]/sigma[1]^2
    alpha[3]=mu[2]*beta[2]-mu[1]*alpha[2]
    if (alpha[3]    }
list(solub=solub,alpha=alpha,beta=beta)
}

mu=runif(2,0,10);sig=c(mu[1]^2/runif(1),mu[2]^2/runif(1));sol(mu,c(sig,runif(1,max(-sqrt(prod(sig)),
-mu[1]*mu[2]),sqrt(prod(sig)))))

and I did not get any FALSE outcome when running this code several times.

What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix?

This is the question I posted this morning on StackOverflow, following an exchange two days ago with a user who could not see why the linear transform of a log-normal vector X,

Y = μ + Σ X

could lead to negative components in Y…. After searching a little while, I could not think of a joint distribution on the positive k-dimensional quadrant where I could specify the covariance matrix in advance. Except for a pedestrian construction of (x₁,x₂) where x₁ would be an arbitrary Gamma variate [with a given variance] and x₂ conditional on x₁ would be a Gamma variate with parameters specified by the covariance matrix. Which does not extend nicely to larger dimensions.