Algebraic Methods toward Higher-order Probability Inequalities, Ii by Donald St. P. Richards

Let (L, ) be a finite distributive lattice, and suppose that the functions f1, f2 :L→ R are monotone increasing with respect to the partial order . Given μ a probability measure on L, denote by E(fi) the average of fi over L with respect to μ, i = 1,2. Then the FKG inequality provides a condition on the measure μ under which the covariance, Cov(f1, f2) := E(f1f2) − E(f1)E(f2), is nonnegative. In this paper we derive a “thirdorder” generalization of the FKG inequality: Let f1, f2 and f3 be nonnegative, monotone increasing functions on L; and let μ be a probability measure satisfying the same hypotheses as in the classical FKG inequality; then