Multivariate Distributions with Fixed Marginals and Correlations

Consider the problem of drawing random variates  ( X 1 ,…, X n )  from a distribution where the marginal of each  X i  is specified, as well as the correlation between every pair  X i  and  X j . For given marginals, the Fr\'echet-Hoeffding bounds put a lower and upper bound on the correlation between  X i  and  X j . Any achievable correlation between  X i  and  X j  is a convex combinations of these bounds. The value  λ ( X i , X j )∈[0,1]  of this convex combination is called here the convexity parameter of  ( X i , X j ),  with  λ ( X i , X j )=1  corresponding to the upper bound and maximal correlation. For given marginal distributions functions  F 1 ,…, F n  of  ( X 1 ,…, X n )  we show that  λ ( X i , X j )= λ ij  if and only if there exist symmetric Bernoulli random variables  ( B 1 ,…, B n )  (that is  {0,1} random variables with mean 1/2) such that  λ ( B i , B j )= λ ij . In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three and four dimensions.