FKG Inequality - A Generalization: The Holley Inequality

A Generalization: The Holley Inequality

The Holley inequality, due to Richard Holley (1974), states that the expectations

of a monotonically increasing function ƒ on a finite distributive lattice with respect to two positive functions μ₁, μ₂ on the lattice satisfy the condition

provided the functions satisfy the Holley condition (criterion)

for all x, y in the lattice.

To recover the FKG inequality: If μ satisfies the lattice condition and ƒ and g are increasing functions on, then μ₁(x)=g(x)μ(x) and μ₂(x)=μ(x) will satisfy the lattice-type condition of the Holley inequality. Then the Holley inequality states that

which is just the FKG inequality.

As for FKG, the Holley inequality follows from the Ahlswede–Daykin inequality.