A BK inequality for randomly drawn subset of fixed size
Journal article, 2012

The BK inequality (van den Berg and Kesten in J Appl Probab 22:556–569, 1985) says that, for product measures on {0, 1} n , the probability that two increasing events A and B ‘occur disjointly’ is at most the product of the two individual probabilities. The conjecture in van den Berg and Kesten (1985) that this holds for all events was proved by Reimer (Combin Probab Comput 9:27–32, 2000). Several other problems in this area remained open. For instance, although it is easy to see that non-product measures cannot satisfy the above inequality for all events, there are several such measures which, intuitively, should satisfy the inequality for all increasing events. One of the most natural candidates is the measure assigning equal probabilities to all configurations with exactly k 1’s (and probability 0 to all other configurations). The main contribution of this paper is a proof for these measures. We also point out how our result extends to weighted versions of these measures, and to products of such measures.

negative dependence

BK inequality


Jacob van den Berg

Vrije Universiteit Amsterdam

Johan Jonasson

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Probability Theory and Related Fields

0178-8051 (ISSN) 1432-2064 (eISSN)

Vol. 154 3-4 835-844

Areas of Advance

Information and Communication Technology


Basic sciences

Subject Categories

Probability Theory and Statistics



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3/6/2018 9