Multivariate distributions with fixed marginals and correlations

Abstract

Consider the problem of drawing random variates $(X_1,\ldots,X_n)$ 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 $\lambda(X_i,X_j) \in [0,1]$ of this convex combination is called here the convexity parameter of $(X_i,X_j),$ with $\lambda(X_i,X_j) = 1$ corresponding to the upper bound and maximal correlation. For given marginal distributions functions $F_1,\ldots,F_n$ of $(X_1,\ldots,X_n)$ we show that $\lambda(X_i,X_j) = \lambda_{ij}$ if and only if there exist symmetric Bernoulli random variables $(B_1,\ldots,B_n)$ (that is $\{0,1\}$ random variables with mean 1/2) such that $\lambda(B_i,B_j) = \lambda_{ij}$. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three and four dimensions.