Algebraic methods toward higher-order probability inequalities, II

Richards, Donald St. P.

Ann. Probab., Tome 32 (2004) no. 1A, p. 1509-1544 / Harvested from Project Euclid

Résumé

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 f_i 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.

Détail

Publié le : 2004-04-14
Classification:  Association,  conjugate cumulants,  diffusion equations,  FKG inequality,  graph theory,  interacting particle systems,  Ising models,  likelihood ratio test statistics,  monotone power functions,  observational studies,  partial orders,  percolation,  Ramsey theory,  reliability theory,  Selberg’s trace formula,  Sperner theory,  total positivity,  60E15,  60J60