The FKG Inequality and Some Monotonicity Properties of Partial Orders
Penn collection
Degree type
Discipline
Subject
Statistics and Probability
Funder
Grant number
License
Copyright date
Distributor
Related resources
Author
Contributor
Abstract
Let (a1 , . . . , am, b1, . . . , bn) be a random permutation of 1, 2, . . ., m + n. Let P be a partial order on the a’s and b’s involving only inequalities of the form ai < aj or bi < bj, and let P' be an extension of P to include inequalities of the form ai < bj; i.e, P' = P ∪ P'', where P'' involves only inequalities of the form ai < bj. We prove the natural conjecture of R. L. Graham, A. C. Yao, and F. F. Yao [SIAM J. Alg. Discr. Meth. 1 (1980), pp. 251–258] that in particular () Pr (a1 < b1|P') ≥ Pr (a1 < b1|P). We give a simple example to show that the more general inequality () where P is allowed to contain inequalities of the form ai < bj is false. This is surprising because as Graham, Yao, and Yao proved, the general inequality (*) does hold if P totally orders both the a’s and the b’s separately. We give a new proof of the latter result. Our proofs are based on the FKG inequality.