Algebraic methods toward higher-order probability inequalities, II

• 成果类型: Article
• 署名作者: Richards, DS
• 署名单位: Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000298
• 发表日期: 2004
• 页码: 1509-1544
• 关键词: holley-preston inequalities; monotonicity properties; positive correlations; fkg inequalities; conjecture; matrices; FORMULA; sets

摘要

Let (L,less than or equal to) 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 less than or equal to. Given mu a probability measure on L, denote by E(f(i)) the average of f(i) over L with respect to mu, i = 1, 2. Then the FKG inequality provides a condition on the measure g 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 mu 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)equivalent to1. 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.