Matching marginals and sums

For a given set of random variables $X_1,\dots ,X_d$ we seek as large a family as possible of random variables $Y_1,\dots ,Y_d$ such that the marginal laws and the laws of the sums match: $Y_i\frac{\text{buildrel}d}{=}{X}_i$ and ${\sum }_i{Y}_i\frac{\text{buildrel}d}{=}{\sum }_i{X}_i$. Under the assumption that $X_1,\dots ,X_d$ are independent and belong to any of the Meixner classes, we give a full characterisation of the random variables $Y_1,\dots ,Y_d$ and propose a practical construction by means of a finite mean square expansion. When $X_1,\dots ,X_d$ 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 $g$, $g(Y)\frac{\text{buildrel}d}{=}g(X)$.