Matching marginals and sums

TL;DR

This paper characterizes and constructs families of random variables matching given marginals and sum distributions, especially for Meixner class variables, with applications to symmetric functions and dependence structures.

Contribution

It provides a full characterization and practical construction methods for matching marginals and sums for Meixner class variables, including symmetric cases.

Findings

Complete characterization for independent Meixner variables

Practical finite mean square expansion construction

Universal symmetric construction for identically distributed variables

Abstract

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