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Let (L,\preccurlyeq) be a finite distributive lattice, and suppose that thefunctions f_1,f_2:L\to R are monotone increasing with respect to the partialorder \preccurlyeq. Given \mu a probability measure on L, denote by E(f_i) theaverage 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 thecovariance, Cov(f_1,f_2):=E(f_1f_2)-E(f_1)E(f_2), is nonnegative. In this paperwe derive a ``third-order'' generalization of the FKG inequality.  We also establish fourth- and fifth-order generalizations of the FKGinequality and formulate a conjecture for a general mth-order generalization.For functions and measures on R^n we establish these inequalities by extendingthe method of diffusion processes. We provide several applications of thethird-order inequality, generalizing earlier applications of the FKGinequality.  Finally, we remark on some connections between the theory of total positivityand the existence of inequalities of FKG-type within the context of Riemannianmanifolds.

Algebraic methods toward higher-order probability inequalities, II