Mathematics – Probability
Scientific paper
2004-10-06
Annals of Probability 2004, Vol. 32, No. 2, 1509-1544
Mathematics
Probability
Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imsta
Scientific paper
10.1214/009117904000000298
Let (L,\preccurlyeq) be a finite distributive lattice, and suppose that the functions f_1,f_2:L\to R are monotone increasing with respect to the partial order \preccurlyeq. 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 \mu under which the covariance, Cov(f_1,f_2):=E(f_1f_2)-E(f_1)E(f_2), is nonnegative. In this paper we derive a ``third-order'' generalization of the FKG inequality. 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.
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