  In this paper we present a method for exact generation of multivariate
samples with pre-specified marginal distributions and a given correlation
matrix, based on a mixture of Fr\'echet-Hoeffding bounds and marginal products.
The bivariate algorithm can accommodate any among the theoretically possible
correlation coefficients, and explicitly provides a connection between
simulation and the minimum correlation attainable for different distribution
families. We calculate the minimum correlations in several common
distributional examples, including in some that have not been looked at before.
As an illustration, we provide the details and results of implementing the
algorithm for generating three-dimensional negatively and positively correlated
Beta random variables, making it the only non-copula algorithm for correlated
Beta simulation in dimensions greater than two. This work has potential for
impact in a variety of fields where simulation of multivariate stochastic
components is desired.
