Algebraic methods toward higher-order probability inequalities, II

Let (L, succeeds or equal to sign) be a finite distributive lattice, and suppose that the functions f ₁, f ₂ : L → ℝ are monotone increasing with respect to the partial order succeeds or equal to sign. Given μ a probability measure on L, denote by E(f _i,-) the average of f _iover L with respect to μ, i = 1, 2. Then the FKG inequality provides a condition on the measure μ under which the covariance, Cov(f ₁, f ₂) := E(f ₁ f ₂) - E(f ₁)E(f ₂), is nonnegative. In this paper we derive a "third-order" generalization of the FKG inequality: Let f ₁, f ₂ and f ₃ 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 2E(f ₁ f ₂ f ₃) - [E(f ₁ f ₂)E(f ₃) + E(f ₁ f ₃)E(f ₂) + E(f ₁)E(f ₂ f ₃)] + E(f ₁)E(f ₂)E(f ₃) is nonnegative. This result reduces to the FKG inequality for the case in which f ₃ ≡ 1. We also establish fourth- and fifth-order generalizations of the FKG inequality and formulate a conjecture for a general mth-order generalization. For functions and measures on ℝ ⁿ we establish these inequalities by extending the method of diffusion processes. We provide several applications of the third-order inequality, generalizing earlier applications of the FKG inequality. Finally, we remark on some connections between the theory of total positivity and the existence of inequalities of FKG-type within the context of Riemannian manifolds.