Pairwise optimal coupling of multiple random variables

TL;DR

This paper extends the optimal coupling theorem to multiple random variables, providing a method to couple them with disagreement probabilities close to their total variation distances, and proves the optimality of these bounds.

Contribution

It introduces a generalized coupling theorem for multiple variables and establishes the sharpness of the bounds using new extremal combinatorics results.

Findings

Coupling multiple variables with disagreement probability close to total variation distance.

Proved the optimality of the coupling bounds in several cases.

Developed new extremal combinatorics results relevant to the problem.

Abstract

We generalize the optimal coupling theorem to multiple random variables: Given a collection of random variables, it is possible to couple all of them so that any two differ with probability comparable to the total-variation distance between them. In a number of cases we show that the disagreement probability we achieve is the best possible. The proofs of sharpness rely on new results in extremal combinatorics, which may be of independent interest.