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A BK inequality for randomly drawn subsets of fixed size
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  • Published: 16 September 2011

A BK inequality for randomly drawn subsets of fixed size

  • J. van den Berg1 &
  • J. Jonasson2 

Probability Theory and Related Fields volume 154, pages 835–844 (2012)Cite this article

Abstract

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.

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Authors and Affiliations

  1. CWI and VU University Amsterdam, Amsterdam, The Netherlands

    J. van den Berg

  2. Chalmers University of Technology and Gothenburg University, Göteborg, Sweden

    J. Jonasson

Authors
  1. J. van den Berg
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  2. J. Jonasson
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Correspondence to J. van den Berg.

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Cite this article

van den Berg, J., Jonasson, J. A BK inequality for randomly drawn subsets of fixed size. Probab. Theory Relat. Fields 154, 835–844 (2012). https://doi.org/10.1007/s00440-011-0386-z

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  • Received: 20 May 2011

  • Revised: 21 July 2011

  • Published: 16 September 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0386-z

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Keywords

  • BK inequality
  • Negative dependence

Mathematics Subject Classification (2000)

  • 60C05
  • 60K35
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