Couplings and Matchings: Combinatorial notes on Strassen's theorem

TL;DR

This paper explores the deep connections between Hall's marriage theorem and Strassen's theorem on couplings, providing new combinatorial proofs and a unifying lemma that links these fundamental results in combinatorics and probability theory.

Contribution

It introduces a novel combinatorial lemma that enables deriving both Hall's and Strassen's theorems, establishing their equivalence in a finite setting with new combinatorial proofs.

Findings

Established the equivalence between Hall's theorem and Strassen's theorem in finite cases.

Provided a new combinatorial lemma for deriving these theorems.

Offered combinatorial proofs previously lacking in the literature.

Abstract

Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall's marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory, in the sense that these results can easily be derived from each other. In this paper it is shown that this equivalence extends to a finite version of Strassen's theorem, a celebrated result on couplings of probability measures. Though this equivalence is known, probabilistic or combinatorial proofs of this fact are lacking. A novel combinatorial lemma will be introduced that can be used to deduce both Hall's and Strassen's theorems.