Simulation of multivariate distributions with fixed marginals and correlations

Consider the problem of drawing random variates (X1, . . . , Xn) from a distribution where the marginal of each Xi is specified, as well as the correlation between every pair Xi and Xj . For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between Xi and Xj . Hence any achievable correlation can be uniquely represented by a convexity parameter λij ∈ [0, 1] where 1 gives the maximum correlation and 0 the minimum correlation. We show that for a given convexity parameter matrix, the worst case is when the marginal distribution are all Bernoulli random variables with parameter 1/2 (fair 0-1 coins). It is worst case in the sense that given a convexity parameter matrix that is obtainable when the marginals are all fair 0-1 coins, it is possible to simulate from any marginals with the same convexity parameter matrix. In addition, we characterize completely the set of convexity parameter matrices for symmetric Bernoulli marginals in two, three and four dimensions.