The inequality conjectured by van den Berg and Kesten in [9], and proved by Reimer in [6], states that for A and B events on S, a product of finitely many finite sets, and P any product measure on S,P(AÊB) £ P(A)P(B), where AÊB are 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 Si (such as Rd), and set S = Õ Si. Say that the function f : S ® R depends on K Í {1,...,n} if f(x) = f(y) whenever xi = yi for all i Î K. Then for any given finite or countable collections of non-negative real valued functions {fa}aÎA, {gb}bÎB on S which depend on Ka and Lb respectively, E{supKaÇLb fa(X) gb(X)} £ E{sup fa(X)} E{sup gb(X)}. Related formulations, and functional versions of the dual inequality on events by Kahn, Saks, and Smyth [4], are also considered. Applications include order statistics, assignment problems, and paths in random graphs.
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Paper provided by Center for Rationality and Interactive Decision Theory, Hebrew University, Jerusalem in its series Discussion Paper Series with number
dp374.
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