A universal approach to matching marginals and sums

For a given set of random variables X₁, …, X_d we seek as large a family as possible of random variables Y₁, …, Y_d such that the marginal laws and the laws of the sums match: (Formula Presented) and (Formula Presented)^.Under the assumption that X₁, …, X_d are identically distributed but not necessarily independent, using a symmetry-balancing approach we provide a universal construction with sufficient symmetry to satisfy the more stringent requirement that, for any symmetric function g, (Formula Presented). The same ideas are shown to extend to the non-identically but “similarly” distributed case.