Algebraic methods toward higher-order probability inequalities, II

Abstract

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 $\mathbb{E}(f_{i})$ 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, $\operatorname{Cov}(f_{1},f_{2}):=\mathbb{E}(f_{1}f_{2})-\mathbb{E}(f_{1})\mathbb{E}(f_{2})$ , is nonnegative. In this paper we derive a “third-order” 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 $$\begin{array}{l}2\mathbb{E}(f_{1}f_{2}f_{3})\\\qquad{}-[\mathbb{E}(f_{1}f_{2})\mathbb{E}(f_{3})+\mathbb{E}(f_{1}f_{3})\mathbb{E}(f_{2})+\mathbb{E}(f_{1})\mathbb{E}(f_{2}f_{3})]\\\qquad{}+\mathbb{E}(f_{1})\mathbb{E}(f_{2})\mathbb{E}(f_{3})\end{array}$$ is nonnegative. This result reduces to the FKG inequality for the case in which f3≡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 ℝn 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.