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.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty