There is a growing need for the ability to model and generate samples of dependent random variables as primitive inputs to stochastic models. We consider the case where this dependence is modeled in terms of a partially-specified finite-dimensional random vector. A random vector sampler is commonly required to match a given set of distributions for each of its components (the marginal distributions) and values of their pairwise covariances. The NORTA method, which produces samples via a transformation of a joint-normal random vector sample, is considered the state-of-the-art method for matching this specification. We begin by showing that the NORTA method has certain flaws in its design which limit its applicability.

A covariance matrix is said to be feasible for a given set of marginal distributions if a random vector exists with these properties. We develop a computational tool that can establish the feasibility of (almost) any covariance matrix for a fixed set of marginals. This tool is used to rigorously establish that there are feasible combinations of marginals and covariance matrices that the NORTA method cannot match. We further determine that as the dimension of the random vector increases, this problem rapidly becomes acute, in the sense that NORTA becomes increasingly likely to fail to match feasible specifications. As part of this analysis, we propose a random matrix sampling technique that is possibly of wider interest.

We extend our study along two natural paths. First, we investigate whether NORTA can be modified to approximately match a desired covariance matrix that the original NORTA procedure fails to match. Results show that simple, elegant modifications to the NORTA procedure can help it achieve close approximations to the desired covariance matrix, and these modifications perform well with increasing dimension.

Second, the feasibility testing procedure suggests a random vector sampling technique that can exactly match (almost) any given feasible set of marginals and covariances, i.e., be free of the limitations of NORTA. We develop a strong characterization of the computational effort needed by this new sampling technique. This technique is computationally competitive with NORTA in low to moderate dimensions, while matching the desired covariances exactly.