Say there are $n$ mean-zero random normal variables $\varepsilon_i,...,\varepsilon_n$ with a $n \times n$ covariance matrix $\Sigma$.
I have $n$ mean-zero random normal variables $u_i, ...,u_n$ which are independent from one-another but may have different variances $\sigma_i^2$ (i.e. the covariance matrix $\Sigma_u$ is diagonal).
Is it possible to write every $\varepsilon_i$ as a sum of the variables $u_1,...,u_n$ in such a way that the covariance matrix is $\Sigma$?
(Another way of phrasing the question is: can I construct a set of $n$ arbitrarily correlated random normal variables by summing $n$ independent random normal variables?)
