Abstract
A multivariate symmetric Bernoulli distribution has marginals that are uniform over the pair {0,1}. Consider the problem of sampling from this distribution given a prescribed correlation between each pair of variables. Not all correlation structures can be attained. Here we completely characterize the admissible correlation vectors as those given by convex combinations of simpler distributions. This allows us to bijectively relate the correlations to the well-known CUTn polytope, as well as determine if the correlation is possible through a linear programming formulation.
Highlights
Consider the admissible correlations among n random variables (X1, . . . , Xn) for given marginal distributions
Since the correlation mapping R is affine, the above theorem says that ρ can be a correlation for an n-variate symmetric Bernoulli distribution if and only if it can be written as a convex combination of R(Pv), for v ∈ {0, 1}n
Characterizing R(Bn) via its extreme points naturally raises the same question about the convex set Set of all n-variate symmetric Bernoulli distributions (Bn)
Summary
Consider the admissible correlations among n random variables (X1, . . . , Xn) for given marginal distributions. While this matrix is in the elliptope E3, it cannot be the correlation matrix of three random variables with symmetric Bernoulli marginals. In this paper we give a complete characterization of the correlation matrices for multivariate symmetric Bernoulli distributions by explicitely identifying vertices of the corresponding polytope.
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