Multivariate distributions with fixed marginals and correlations

Abstract

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 betweenXi and Xj. Any achievable correlation between Xi and Xj is a convex combination of these bounds. We call the value λ(Xi, Xj) ∈ [0, 1] of this convex combination the convexity parameter of (Xi, Xj) with λ(Xi, Xj) = 1 corresponding to the upper bound and maXimal correlation. For given marginal distributions functions F1,..., Fn of (X1,..., Xn), we show that λ(Xi, Xj) = λij if and only if there exist symmetric Bernoulli random variables (B1,..., Bn) (that is {0, 1} random variables with mean 1/2) such that λ(Bi, Bj) = λij. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.