A combinatorial approach to correlation inequalities
This paper presents new proofs of two theorems originally proved in the 1980s. Consider an arbitrary finite partially ordered set \(P\). For two elements \(x\) and \(y\) of \(P\), the probability that \(x>y\), \(\text{prob}[x>y]\), is the fraction of linear extensions of \(P\) in which \(x>y\). The first theorem, known as the ``strong XYZ theorem'', states that if \(x\), \(y\) and \(z\) are incomparable elements of \(P\), then the probabilities that \(x>y\) and \(x>z\) are not independent: \(\text{prob}[x>y] \text{prob}[x>z]< \text{prob}[x>y,x>z]\). A weaker version of this was first proved by \textit{L. A. Shepp} [Ann. Probab. 10, 824-827 (1982; Zbl 0484.60010)]. Later, the strong XYZ theorem was proved by \textit{P. C. Fishburn} [Order 1, 127-137 (1984; Zbl 0562.06002)]. The other theorem proved here was first proved by \textit{L. A. Shepp} [SIAM J. Algebraic Discrete Methods 1, 295-299 (1980; Zbl 0501.06006)]. It gives an inequality on the probabilities of certain types of extensions of a poset whose elements can be partitioned into two sets, with each element of one set incomparable to all elements of the other. Unlike the original proofs, the new proofs do not use the Ahlswede-Daykin Four Function Theorem.
scientific article; zbMATH DE number 1838995