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Functional BRK Inequalities, and their Duals, with Applications

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  • Larry Goldstein
  • Yosef Rinott

Abstract

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 R d ), and set S = Õ Si. Say that the function f : S ® R depends on K Í {1,...,n} if f(x) = f(y) whenever x i = y i for all i Î K. Then for any given finite or countable collections of non-negative real valued functions {f a } a Î A , {g b } b Î B on S which depend on K a and L b respectively, E{sup K a Ç L b f a (X) g b (X)} £ E{sup f a (X)} E{sup g b (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.

Suggested Citation

  • Larry Goldstein & Yosef Rinott, 2004. "Functional BRK Inequalities, and their Duals, with Applications," Discussion Paper Series dp374, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
  • Handle: RePEc:huj:dispap:dp374
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    1. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
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    Keywords

    graphs and paths; positive dependence; order statistics;

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