Functional BKR inequalities, and their duals, with applications

The inequality conjectured by van den Berg and Kesten (J. Appl. Probab. 22, 556-569, 1985), and proved by Reimer (Comb. Probab. Comput. 9, 27-32, 2000), states that for A and B events on S, a finite product of finite sets, and P any product measure on S, P(A□B)≤ P(A)P(B), where the set A□ B consists of the elementary events which lie in both A and B for 'disjoint reasons.' This inequality on events is the special case, for indicator functions, of the inequality having the following formulation. Let X be a random vector with n independent components, each in some space S _i (such as R ^d ), and set S= ∏ⁿ _i=1 ^s _i. Say that the function f:S→r depends on K ⊆{1,...n} if f(x)=f(y) whenever x _i =y _i for all i ∞K. Then for any given finite or countable collections of non-negative real valued functions {f α}_{α ∞}A},{gβ}β ∞B on S, depending on K _α,α A and L _{β ∞} B respectively, E{k α∩lpha} L_β=θf _α (X)g_β (X)}≤ E {sup_αf α(X)}E {(sup gβ β(X)}. Related formulations, and functional versions of the dual inequality on events by Kahn, Saks, and Smyth (15th Annual IEEE Conference on Computational Complexity 98-103, IEEE Computer Soc., Los Alamitos, CA, 2000), are also considered. Applications include order statistics, assignment problems, and paths in random graphs.