Let a correlation matrix $\Sigma$ be given. I would like to sample from the n-dimensional multivariate beta distribution where each marginal distribution is known and the variables are correlated as prescribed by $\Sigma$. Is there a way to do this, preferably one that is not very expensive computationally?
There is a lot of info on how to solve similar problems for the normal distribution (see link to Wikipedia with a well-known procedure, or this question or this question) and there are some questions on how to do this for the special case of $n=2$ (for example this). But I cannot find information on this specific question here, neither on Google or SE.