The Annals of Probability

On a Combinatorial Conjecture Concerning Disjoint Occurrences of Events

J. Van Den Berg and U. Fiebig

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Recently van den Berg and Kesten have obtained a correlation-like inequality for Bernoulli sequences. This inequality, which goes in the opposite direction of the FKG inequality, states that the probability that two monotone (i.e., increasing or decreasing) events "occur disjointly" is smaller than the product of the individual probabilities. They conjecture that the monotonicity condition is immaterial, i.e., that the inequality holds for all events. In the present paper we try to make clear the intuitive meaning of the conjecture and prove some nontrivial special cases, one of which, a pure correlation inequality, is an extension of Harris' FKG inequality.

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Ann. Probab., Volume 15, Number 1 (1987), 354-374.

First available in Project Euclid: 19 April 2007

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Primary: 60C05: Combinatorial probability
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Disjoint occurrences of events correlation inequality FKG inequality combinatorial probability percolation finite Bernoulli sequences


Berg, J. Van Den; Fiebig, U. On a Combinatorial Conjecture Concerning Disjoint Occurrences of Events. Ann. Probab. 15 (1987), no. 1, 354--374. doi:10.1214/aop/1176992274.

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