### Abstract

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 _{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, 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.

Original language | English (US) |
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Pages (from-to) | 1509-1544 |

Number of pages | 36 |

Journal | Annals of Probability |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2004 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty