Higher correlation inequalities

The purpose of this paper is to prove a correlation inequality for n increasing functions on a distributive lattice. For finite lattices, it is possible to combine the entire family of inequalities (for all n) into a single statement involving the ring of formal power series. This is the form in which we first formulate and prove our result. Thus, let X be a finite set and regard the power set 2X as a partially ordered set with respect to inclusion. Let R :=R[[t]] be the space of formal power series in the variable t with real coefficients. The set P := {a1t+ a2t + · · · ∈ R | ai ≥ 0 for all i} is a convex cone in R, and we define R [X] := {F | F : 2 → R} and P [X] := {F | F : 2 → P} . Now P [X] is a convex cone in R [X] and we will refer to its elements as positive functions on 2X . We define the subcone of increasing positive functions